2.1 Euclidean norms

The simplest norm considered on $\mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ is the one induced by the standard Euclidean inner product in a monomial basis. Every $f\in \mathcal {H}^{\mathbb {F}}_d[1]$ can be uniquely represented as

(2.1) $$ \begin{align} f=\sum_{|\alpha|=d} f_\alpha X^\alpha, \end{align} $$

where $\alpha =(\alpha _0,\ldots ,\alpha _n)\in {\mathbb {N}}^{n+1}$ and $|\alpha |=\alpha _0+\cdots +\alpha _n$ . The norm induced by the standard Euclidean inner product is therefore

(2.2) $$ \begin{align} \|f\|_{\mathrm{std}}:=\sqrt{\sum_{|\alpha|=d}|f_\alpha|^2}. \end{align} $$

For $f=(f_1,\ldots ,f_q)\in \mathcal {H}_{\boldsymbol {d}}[q]$ , the norm extends as $\|f\|_{\mathrm {std}}^2:=\|f_1\|_{\mathrm {std}}^2+\cdots +\|f_q\|_{\mathrm {std}}^2$ .

The most commonly used norm on $\mathcal {H}_{\boldsymbol {d}}[q]$ is the Weyl norm. For a polynomial as in equation (2.1), this is given by

(2.3) $$ \begin{align} \|f\|_W:=\sqrt{\sum_{|\alpha|=d}\binom{d}{\alpha}^{-1} |f_\alpha|^2}, \end{align} $$

where $\binom {d}{\alpha }$ is the multinomial coefficient $\frac {d!}{\alpha _0!\ldots \alpha _n!}$ . Again, for $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ , this extends by $\|f\|_W^2:=\|f_1\|_W^2+\cdots +\|f_q\|_W^2$ . The Weyl norm is also induced by an inner product, and this inner product is invariant under the action of the unitary group (respectively, the orthogonal group when the underlying field is $\mathbb {R}$ ). It is straightforward to check that, for $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ ,

$$ \begin{align*}\|f\|_W \le \|f\|_{\mathrm{std}}\le \max_{i\le q}\max_{|\alpha|=d_i}\binom{d_i}{\alpha} \|f\|_W. \end{align*} $$

Here, and in all that follows, for any $x\in \mathbb {S}^n$ and $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ , $\mathrm {D}_xf:\mathrm {T}_x\mathbb {S}^n\to \mathbb {R}^q$ is the derivative of f at x restricted to the tangent space $\mathrm {T}_x\mathbb {S}^n$ of $\mathbb {S}^n$ at x. A similar convention applies in the complex case replacing $\mathbb {S}^n$ and $\mathrm {T}_x\mathbb {S}^n$ by $\mathbb {P}^n$ and $\mathrm {T}_{z}\mathbb {P}^n$ . The following property (see [Reference Bürgisser and Cucker14, Proposition 16.16]) is one of the most important properties of the Weyl norm from the viewpoint of the complexity of numerical algorithms.

Proposition 2.2. For all $x\in \mathbb {S}^n$ , the map

$$ \begin{align*} \mathcal{H}_{\boldsymbol{d}}[q]\ni f\mapsto \mathrm{ev}_xf:=\left(f(x),\Delta^{-\frac{1}{2}}\mathrm{D}_x f\right) \end{align*} $$

is an orthogonal projection from $\mathcal {H}_{\boldsymbol {d}}[q]$ endowed with the Weyl norm onto $\mathbb {R}^{q}\times \mathrm {T}_x\mathbb {S}^n\simeq \mathbb {R}^{q+n}$ equipped with the standard Euclidean norm. An analogous statement holds in the complex case.

2.2 Functional norms

We will consider functional norms that arise from evaluating polynomials at points on the sphere. One might consider other norms (as we do in Section A), but $L_p$ -norms suffice for obtaining the computational improvements we aim for. Although in the sequel we will only use the $L_\infty $ -norm, we present the full family of $L_p$ -norms since we consider that these norms will be useful in the future. Moreover, presenting the full family of $L_p$ -norms allows us to appreciate how the $L_\infty $ differs from and relates to these other norms.

We will consider the two following classes of L-norms on $\mathcal {H}_{\boldsymbol {d}}[q]$ :

• $(\mathbb {R})$ Real $L_p$ -norm: For $p \in [1,\infty ]$ ,

  $$ \begin{align*}\|f\|_{p}^{\mathbb{R}}:=\begin{cases} \displaystyle\max_{x\in\mathbb{S}^n}\|f(x)\|_{\infty}=\max_{x\in\mathbb{S}^n}\max_i|f_i(x)|&\text{if }p=\infty\\ \displaystyle\left(\mathop{\mathbb{E}}_{\mathfrak{x}\in\mathbb{S}^n}\|f(\mathfrak{x})\|_p^p\right)^{1/p}=\left(\mathop{\mathbb{E}}_{\mathfrak{x}\in\mathbb{S}^n}\left(\sum_{i=1}^q|f_i(\mathfrak{x})|^p\right)\right)^{1/p}&\text{otherwise},\end{cases}\end{align*} $$
  where the expectations are taken over the uniform distribution of the n-dimensional sphere $\mathbb {S}^n\subseteq \mathbb {R}^{n+1}$ .

• $(\mathbb {C})$ Complex $L_p$ -norm: For $p \in [1,\infty ]$ ,

  $$ \begin{align*}\|f\|_{p}^{\mathbb{C}}:=\begin{cases} \displaystyle\max_{z\in\mathbb{P}^n}\left\|f(z)\right\|_{\infty}=\max_{z\in\mathbb{P}^n}\max_i\left|f_i(z)\right|&\text{if }p=\infty\\ \displaystyle\left(\mathop{\mathbb{E}}_{\mathfrak{z}\in\mathbb{P}^n}\left\|f(\mathfrak{z})\right\|_p^p\right)^{1/p}=\left(\mathop{\mathbb{E}}_{\mathfrak{z}\in\mathbb{P}^n}\left(\sum_{i=1}^q\left|f_i(\mathfrak{z})\right|{}^p\right)\right)^{1/p}&\text{otherwise},\end{cases}\end{align*} $$
  where the expectations are taken over the uniform distribution of the complex n-dimensional projective space $\mathbb {P}^n:=\mathbb {P}^n_{\mathbb {C}}$ .

Remark 2.3. In the case of a single polynomial, the definitions above become simpler. For $f\in \mathcal {H}_{\boldsymbol {d}}[1]$ ,

$$ \begin{align*} \|f\|_{p}^{\mathbb{R}}:=\begin{cases} \displaystyle\max_{x\in\mathbb{S}^n} \left\lvert {f(x)} \right\rvert &\text{if }p=\infty\\ \displaystyle\left(\mathop{\mathbb{E}}_{\mathfrak{x}\in\mathbb{S}^n}|f(\mathfrak{x})|^p\right)^{1/p}&\text{otherwise}\end{cases}~~\text{ and }~~\|f\|_{p}^{\mathbb{C}}:=\begin{cases} \displaystyle\max_{z\in\mathbb{P}^n}\left|f(z)\right|&\text{if }p=\infty\\ \displaystyle\left(\mathop{\mathbb{E}}_{\mathfrak{z}\in\mathbb{P}^n}\left|f(\mathfrak{z})\right|{}_p^p\right)^{1/p}&\text{otherwise},\end{cases} \end{align*} $$

which amount to taking the p-mean of $|f|$ over, respectively, $\mathbb {S}^n$ and $\mathbb {P}^n$ .

In general, we will omit the superscript when the context is clear. It will be common for us to work with the norms $\|~\|_{p}^{\mathbb {R}}$ in $\mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and the norms $\|~\|_{p}^{\mathbb {C}}$ in $\mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ .Footnote ¹

Our definition has some arbitrary choices. These are motivated by the following two properties:

1. (D) For $p\in [1,\infty ]$ and $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ ,

   $$ \begin{align*}\|f\|_p^{\mathbb{R}}=\left\|\left(\|f_1\|_p^{\mathbb{R}},\ldots,\|f_q\|_p^{\mathbb{R}}\right)\right\|_p~~\text{and}~~\|f\|_p^{\mathbb{C}} =\left\|\left(\|f_1\|_p^{\mathbb{C}},\ldots,\|f_q\|_p^{\mathbb{C}}\right)\right\|_p.\end{align*} $$

   This identity is why we take the p-mean of the p-norm of $f(x)$ instead of taking the p-mean of a fixed norm.

2. (I) We have actions of the qth power of the (real) orthogonal group, $\mathscr {O}(n+1)^q$ , on $\mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ , given by $(A,f)\mapsto (f_i^{A_i}):=(f_i(A_iX))$ . Similarly, we have an action of the qth power of the unitary group, $\mathscr {U}(n+1)^q$ , on $\mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ . The norms $\|~\|_{p}^{\mathbb {R}}$ and $\|~\|_{p}^{\mathbb {C}}$ are invariant under these actions.

We perform some simple computations to have a better grasp of the introduced norms.

Example 2.4 (Monomials).

We consider the value of the norms for a monomial $X^{\alpha }\in \mathcal {H}_d[1]$ of degree d. In this case, we have that for $p\in [1,\infty )$ ,

$$ \begin{align*} \left\|X^\alpha\right\|_p^{\mathbb{R}}=\left(\frac{\Gamma\left(\frac{n+1}{2}\right)\prod_{i=0}^n\Gamma\left(\frac{p\alpha_i+1}{2}\right)}{\pi^{\frac{n+1}{2}}\Gamma\left(\frac{pd+n+1}{2}\right)}\right)^{\frac{1}{p}}~~\text{and}~~ \left\|X^\alpha\right\|_p^{\mathbb{C}}=\left(n!\frac{\prod_{i=0}^n\Gamma\left(\frac{p\alpha_i}{2}+1\right)}{\Gamma\left(\frac{pd}{2}+n+1\right)}\right)^{\frac{1}{p}}, \end{align*} $$

where $\Gamma $ is Euler’s Gamma function, and that

$$ \begin{align*}\left\|X^\alpha\right\|_\infty^{\mathbb{R}}=\left\|X^\alpha\right\|_\infty^{\mathbb{C}}=\prod_{i=0}^n\left(\frac{\alpha_i}{d}\right)^{\frac{\alpha_i}{2}}=\sqrt{\frac{1}{d^d}\prod_{i=0}^n\alpha_i^{\alpha_i}}.\end{align*} $$

For the calculations of $L_p$ -norms of monomials, we refer the reader to [Reference Folland36]. Although the calculation is only illustrated over the reals in the reference, the complex case is similar. For the second one, note that for monomials, real and complex $\infty $ -norms are equivalent. Once this is clear, we are just using the method of Lagrange multipliers to compute the maximum over the sphere.

Example 2.5 (Linear functions).

Let and . Then f can be identified with a matrix A of size $q\times (n+1)$ . We can see that

$$ \begin{align*}\|f\|_{\infty}=\|A\|_{2,\infty}:= \sup_{x\neq0}\frac{\|Ax\|_{\infty}}{\|x\|_2},\end{align*} $$

where $\|~\|_{2,\infty }$ is the operator norm, where the domain vector space has the usual Euclidean norm $\|~\|_2$ and the codomain the $\infty $ -norm $\|~\|_{\infty }$ .

For $p\in [1,\infty )$ ,

$$ \begin{align*} \|f\|_p^{\mathbb{R}}=\|X_0\|_p^{\mathbb{R}}\left\|\left(\|A^1\|_2,\ldots,\|A^q\|_2\right)\right\|_p ~~\text{and}~~ \|f\|_p^{\mathbb{C}}=\|X_0\|_p^{\mathbb{C}}\left\|\left(\|A^1\|_2,\ldots,\|A^q\|_2\right)\right\|_p, \end{align*} $$

where $A^i$ is the ith row of A and $X_0$ is a variable (and hence $\|X_0\|^{\mathbb {F}}_p$ is given by the expressions in Example 2.4). Note that $\left \|\left (\|A^1\|_2,\ldots ,\|A^q\|_2\right )\right \|_p$ is just the p-norm of the vector of $2$ -norms of the rows of A.

Example 2.6 (Sum of squares).

Let $f:=\sum _{i=0}^nX_i^2\in \mathcal {H}_{2}[1]$ . As this function is constant on the real sphere, we have that for all $p\in [1,\infty ]$ ,

$$ \begin{align*}\|f\|_p^{\mathbb{R}}=1.\end{align*} $$

However, on $\mathbb {P}^n$ , f does not behave as a constant function as it has a positive dimensional zero set. Again, arguing as in [Reference Folland36], we can conclude that

$$ \begin{align*} \|f\|_p^{\mathbb{C}}=\left(\frac{1}{\pi^{n+1}}\frac{n!}{(n+p)!}\int_{z\in\mathbb{C}^{n+1}}|f(z)|^p\mathrm{e}^{-|z|^2}\right)^{\frac{1}{p}} \end{align*} $$

for $p\in [1,\infty )$ . Now, if p is even, we can obtain the expression

$$ \begin{align*} \|f\|_p^{\mathbb{C}}=\left(\binom{n+2}{2}^{-1}\sum_{\substack{\alpha\in\mathbb{N}^{n+1}\\|\alpha|=p/2}}\binom{p/2}{\alpha}^2\binom{p}{2\alpha}^{-1}\right)^{\frac{1}{p}} \end{align*} $$

after writing $|f(z)|^p=f(z)^{\frac {p}{2}}\overline {f(z)}^{\kern1pt\frac {p}{2}}$ , expanding and using separation of variables. In particular, for $p=2$ , we obtain that

$$ \begin{align*} \|f\|_2^{\mathbb{C}}=\sqrt{\frac{2}{n+2}}\neq 1. \end{align*} $$

This shows how the norms $\|~\|_p^{\mathbb {C}}$ may be smaller than their corresponding norm $\|~\|_p^{\mathbb {R}}$ for $p\in [1,\infty )$ .

Example 2.7 (Cosine polynomials).

Let $d\geq 2$ , and consider the family of homogeneous polynomials

$$ \begin{align*}c_d:=\sum_{k=0}^{\lfloor d/2\rfloor}\binom{d}{2k}(-1)^kX^{d-2k}Y^{2k}=\frac{1}{2}(X+iY)^d+\frac{1}{2}(X-iY)^d\in\mathcal{H}_d[1].\end{align*} $$

Since $c_d(\cos \theta ,\sin \theta )=\cos d\theta $ , we have that

$$ \begin{align*}\|c_d\|_{\infty}^{\mathbb{R}}=1.\end{align*} $$

Also, $c_d$ is unitarily equivalent to $2^{\frac {d}{2}-1}(X^d+Y^d)$ . Hence

$$ \begin{align*}\|c_d\|_{\infty}^{\mathbb{C}}=2^{\frac{d}{2}-1},\end{align*} $$

since $\|X^d+Y^d\|_{\infty }^{\mathbb {C}}=1$ for $d\geq 2$ . This shows that for degrees $d\geq 3$ , the norms $\|~\|_{\infty }^{\mathbb {R}}$ and $\|~\|_{\infty }^{\mathbb {C}}$ disagree on real polynomials.

The following proposition lists simple inequalities between the functional norms. For a converse of some of the inequalities below, where the $L_\infty $ norm is bounded in terms of $L_p$ norms, see [Reference Barvinok6].

Proposition 2.8. Let $1\leq p < p'<\infty $ and $\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$ . Then for all $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {F}}[q]$ , the following inequalities hold:

$$ \begin{align*}\frac{1}{q^{\frac{1}{p}}}\|f\|_p^{\mathbb{F}}\leq \frac{1}{q^{\frac{1}{p'}}}\|f\|_{p'}^{\mathbb{F}}\leq \|f\|_{\infty}^{\mathbb{F}}\leq \|f\|_{\infty}^{\mathbb{C}}.\end{align*} $$

The Weyl norm is essentially a scaled version of the complex $L_2$ norm.

Proposition 2.9. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ . Then

$$ \begin{align*}\|f\|_W=\sqrt{\sum_{i=1}^qN_i \left(\|f_i\|_{2}^{\mathbb{C}}\right)^2}.\end{align*} $$

In particular, for $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[1]$ ,

$$ \begin{align*} \|f\|^{\mathbb{C}}_W=\sqrt{N}\|f\|_2^{\mathbb{C}}. \end{align*} $$

From Proposition 2.2, we get the following result.

Proposition 2.10. Let $\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$ and $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ . Then for all $p\geq 2$ ,

$$ \begin{align*}\|f\|_p^{\mathbb{F}}\leq \|f\|_W.\end{align*} $$

We finish this subsection by noting how the $L_\infty $ -norms relate to the Weyl norm. We note that this is related to the so-called best rank-one approximation of a symmetric tensor [Reference Agrachev, Kozhasov and Uschmajew1, Reference Zhang, Ling and Qi59]; the inequality for the real case below was already present in [Reference Zhang, Ling and Qi59, Theorem 2.4].

Proposition 2.11. Let $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ . Then

$$ \begin{align*}\|f\|_\infty^{\mathbb{C}}\leq \|f\|_W\leq \sqrt{N}\|f\|_\infty^{\mathbb{C}}.\end{align*} $$

If $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ , then

$$ \begin{align*}\|f\|_\infty^{\mathbb{R}}\leq \|f\|_W\leq (n+1)^{\frac{\mathbf{D}}{2}}\|f\|_\infty^{\mathbb{R}}.\end{align*} $$

Example 2.12. Proposition 2.11 is almost optimal for $n=1$ . In [Reference Agrachev, Kozhasov and Uschmajew1], it was shown that for the cosine polynomials $c_d$ of Example 2.7, we have

$$ \begin{align*}\|c_d\|_W=2^{\frac{d-1}{2}}\end{align*} $$

and that $c_d$ is the real polynomial of real $L_\infty $ norm 1 with largest Weyl norm. Curiously, in this case, the Weyl norm and the complex $L_\infty $ are almost equal, the former being the latter times $\sqrt {2}$ .

2.3 Kellogg’s theorem

We will denote by $\overline {\mathrm {D}}$ the operation of taking all partial derivatives with respect to all variables: that is, $f\mapsto \overline {\mathrm {D}} f$ is a linear map , and for $x\in \mathbb {F}^{n+1}$ , $\overline {\mathrm {D}}_x f:\mathbb {F}^{n+1}\to \mathbb {F}^q$ is a linear map. We will write $\overline {\mathrm {D}}_Xf$ , with a capital X, to emphasise that we view $\overline {\mathrm {D}}_Xf$ as a polynomial tuple in ; and we will write $\overline {\mathrm {D}}_xf$ , with a lowercase x, to emphasise that we view $\overline {\mathrm {D}}_xf$ as the linear map $\mathbb {F}^{n+1}\rightarrow \mathbb {F}^q$ defined at the point x. We also recall that $\mathrm {D}_xf$ is the tangent map $\mathrm {T}_x\mathbb {S}^n\rightarrow \mathbb {R}^q$ in the real case and the tangent map $\mathrm {T}_x\mathbb {P}^n\rightarrow \mathbb {C}^q$ in the complex case.

The following result plays the role of Proposition 2.2 for the infinity norm instead of the Weyl one. It is a reformulation of a well-known inequality proved in [Reference Kellogg40].

Theorem 2.13 (Kellogg’s inequality).

Let $\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$ , $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {F}}[q]$ and $v\in \mathbb {F}^{n+1}$ ; then

$$ \begin{align*}\left\|\Delta^{-1}\overline{\mathrm{D}}_X f v\right\|_\infty^{\mathbb{F}} \leq \|f\|_\infty^{\mathbb{F}}\|v\|.\end{align*} $$

Corollary 2.14. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {F}}[q]$ and $z\in \mathbb {S}^n$ (if $\mathbb {F}=\mathbb {R}$ ) or $z\in \mathbb {P}^n$ (if $\mathbb {F}=\mathbb {C}$ ). Then

$$ \begin{align*} \max\left\{\left\|f(z)\right\|_{\infty},\left\|\Delta^{-1}\mathrm{D}_z f\right\|_{2,\infty}\right\} \leq \|f\|_\infty^{\mathbb{F}}. \end{align*} $$

Before proving Theorem 2.13 and Corollary 2.14, we discuss some features of these results.

Remark 2.15. We note that the left-hand side in Corollary 2.14 is not optimal. In general, we have that

$$ \begin{align*}\|\Delta^{-1}\overline{\mathrm{D}}_xf\|_{2,\infty}=\max_i \sqrt{|f_i(x)|^2+\frac{1}{d_i^2}\|\mathrm{D}_xf_i\|_{2,\infty}^2}.\end{align*} $$

The following examples show how the bound of Theorem 2.13 looks in a few particular cases.

Example 2.16. Consider the cosine polynomials $c_d$ of Example 2.7. A direct computation shows that

$$ \begin{align*}\frac{1}{d}\overline{\mathrm{D}}_Xc_d\,v=v_Xc_{d-1}-v_Ys_{d-1},\end{align*} $$

where $s_{d-1}:=-\frac {i}{2}(X+iY)^{d-1}+\frac {i}{2}(X-iY)$ is the sine polynomial for which $s_{d}(\cos \theta ,\sin \theta )=\sin d\theta $ .

In the real case, this gives

$$ \begin{align*}\left\|\frac{1}{d}\overline{\mathrm{D}}_Xc_dv\right\|_{\infty}^{\mathbb{R}}=\|v\|_{2}=\|c_d\|_\infty^{\mathbb{R}}\|v\|_{2},\end{align*} $$

using the Cauchy-Schwarz inequality. In the complex case, $\frac {1}{d}\mathrm {D}_Xc_dv=v_Xc_{d-1}-v_Ys_{d-1}$ is unitarily equivalent to

$$ \begin{align*}\frac{2^{\frac{d-1}{2}}}{d}\left[(v_{X}-iv_Y) X^{d-1}+(v_{X}+iv_Y)Y^{d-1}\right].\end{align*} $$

Now, $\left \|(v_{X}-iv_Y)x^{d-1} +(v_{X}+iv_Y)y^{d-1}\right \| \leq \sqrt {2} \|v\|_{2}(|x|^{d-1}+|y|^{d-1}) \leq \|v\|_{2}$ for $d\leq 3$ , and v real when $|x|^2+|y|^2\leq 1$ . Thus

$$ \begin{align*}\left\|\frac{1}{d}\overline{\mathrm{D}}_Xc_dv\right\|_{\infty}^{\mathbb{C}}=\frac{2^d}{d}\|v\|_{2}=\frac{\sqrt{2}}{d}\|c_d\|_\infty^{\mathbb{C}}\|v\|_{2}.\end{align*} $$

This shows that the real version of Kellogg’s theorem is tight for $c_d$ , but the complex version is not.

Example 2.17. The reverse situation is true for the polynomial $X_{0}^d$ . One can see that

$$ \begin{align*}\left\|\frac{1}{d}\overline{\mathrm{D}}_X X_{0}^d e_{0}\right\|_\infty^{\mathbb{C}} =\|X_{0}^d\|_\infty^{\mathbb{C}}.\end{align*} $$

Now it is the complex Kellogg’s theorem that is tight. We note, however, that one might still improve Corollary 2.14. For example, is it possible to substitute $\Delta $ by $\Delta ^{\frac {1}{2}}$ in this corollary?

Proof of Theorem 2.13.

We only prove the real case. The complex case is proven analogously (see [Reference Kellogg40, Section 8] for the complex version of the results we use in the real case).

By [Reference Kellogg40, Theorem IV], we have that for all i and all $x\in \mathbb {S}^n$ ,

$$ \begin{align*}\left|\overline{\mathrm{D}}_xf_i v\right|\leq d_i\|f_i\|_{\infty}^{\mathbb{R}}\|v\|,\end{align*} $$

since $\overline {\mathrm {D}}_xf_iv$ is the directional derivative of f at x in the direction of v. Therefore, for all $x\in \mathbb {S}^n$ ,

$$ \begin{align*}\left\|\Delta^{-1}\overline{\mathrm{D}}_xfv\right\|_{\infty}=\max_i \frac{1}{d_i}\left|\overline{\mathrm{D}}_xfv\right|\leq \max_i\|f_i\|_{\infty}^{\mathbb{R}}\|v\| =\|f\|_\infty^{\mathbb{R}}\|v\|.\end{align*} $$

Now $\left \|\Delta ^{-1}\overline {\mathrm {D}}_{X}fv\right \|_{\infty }^{\mathbb {R}} =\max _{x\in \mathbb {S}^n}\|\Delta ^{-1}\overline {\mathrm {D}}_xfv\|_{\infty }$ by definition of $\|~\|_{\infty }^{\mathbb {R}}$ , so we are done.

Proof of Corollary 2.14.

We only prove the real case, the proof for the complex case being essentially the same. Recall that by Euler’s formula for homogeneous functions,

(2.4) $$ \begin{align} \Delta^{-1}\overline{\mathrm{D}}_xfx=f(x). \end{align} $$

In this way, for $x\in \mathbb {S}^n$ , $\lambda \in \mathbb {R}$ and $w\in \mathrm {T}_x\mathbb {S}^n=x^{\perp }$ ,

$$ \begin{align*}\Delta^{-1}\overline{\mathrm{D}}_xf(\lambda x+w)=\lambda f(x)+\Delta^{-1}\mathrm{D}_xfw.\end{align*} $$

When $\lambda x+w=x$ , this expression yields $f(x)$ ; and when $\lambda x+w=w$ , it yields $\Delta ^{-1}\mathrm {D}_xfw$ . In this way,

$$ \begin{align*}\max_{\lambda x+w\neq 0}\frac{\|\Delta^{-1}\overline{\mathrm{D}}_xf(\lambda x+w)\|_{\infty}}{\sqrt{|\lambda|^2+\|w\|^2}}\geq \max\left\{\|f(x)\|_\infty,\max_{v\in\mathrm{T}_x\mathbb{S}^n\setminus 0}\frac{\|\Delta^{-1}\mathrm{D}_xv\|_{\infty}}{\|v\|}\right\}.\end{align*} $$

The left-hand side is bounded by $\|f\|_\infty ^{\mathbb {R}}$ by Theorem 2.13, and the right-hand side equals $\max \{\|f(x)\|_{\infty },\|\Delta ^{-1}\mathrm {D}_xf\|_{2,\infty }\}$ . Thus the desired inequality follows.

Following the notations introduced above, we will write $\overline {\mathrm {D}}^k_xf$ to denote the kth derivative map of $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ at $x\in \mathbb {F}^{n+1}$ . This is the k-multilinear map $(\mathbb {F}^{n+1})^k\rightarrow \mathbb {F}^q$ given by the kth derivatives of f at x. Also, $\overline {\mathrm {D}}^k_Xf(v_1,\ldots ,v_k)$ , where $v_1,\ldots ,v_k\in \mathbb {F}^{n+1}$ will denote the corresponding polynomial tuple in . For a real k-multilinear map $A:(\mathbb {R}^n)^k\rightarrow \mathbb {R}^q$ , we define

(2.5) $$ \begin{align} \|A\|_{2,\infty}^{\mathbb{R}}:=\sup_{v_1,\ldots,v_k\neq 0}\frac{\|A(v_1,\ldots,v_k)\|_\infty}{\|v_1\|\cdots\|v_k\|}. \end{align} $$

We define $\|A\|_{2,\infty }^{\mathbb {C}}$ for a complex k-multilinear map $A:(\mathbb {C}^n)^k\rightarrow \mathbb {C}^q$ in a similar manner. Note that for $k>2$ , by the following corollary and Example 2.7,

$$ \begin{align*}\left\|\frac{1}{k!}\overline{\mathrm{D}}_{0}^kc_k\right\|_{2,\infty}^{\mathbb{C}} =\left\|c_k\right\|_\infty^{\mathbb{C}}=2^{\frac{k}{2}-1}>1 =\left\|c_k\right\|_\infty^{\mathbb{R}} =\left\|\frac{1}{k!}\overline{\mathrm{D}}_{0}^kc_k\right\|_{2,\infty}^{\mathbb{R}}, \end{align*} $$

so for real A, $\|A\|_{2,\infty }^{\mathbb {R}}$ and $\|A\|_{2,\infty }^{\mathbb {C}}$ are not necessarily equal and can differ by a factor exponential in k. The following corollary (which is closely related to [Reference Zhang, Ling and Qi59, Theorem 2.1]) will be useful later.

Corollary 2.20. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {F}}[q]$ and $z\in \mathbb {S}^n$ (if $\mathbb {F}=\mathbb {R}$ ) or $z\in \mathbb {P}^n$ (if $\mathbb {F}=\mathbb {C}$ ). Then for all $k\geq 1$ and $v_1,\ldots ,v_k\in \mathbb {F}^{n+1}$ ,

$$ \begin{align*} \left\|\frac{1}{k!}\Delta^{-1}\overline{\mathrm{D}}_X^kf(v_1,\ldots,v_k)\right\|_{\infty}\leq \frac{1}{k}\binom{\mathbf{D}-1}{k-1}\|f\|_{\infty}^{\mathbb{F}}\|v_1\|\cdots \|v_k\|. \end{align*} $$

In particular, $\left \|\frac {1}{k!}\Delta ^{-1}\overline {\mathrm {D}}_z^kf\right \|_{2,\infty }\leq \frac {1}{k}\binom {\mathbf {D}-1}{k-1}\|f\|_{\infty }^{\mathbb {F}}$ .

3.1 Properties of the real condition number $\mathsf {K}$

Recall (see, e.g., [Reference Bürgisser and Cucker14, Definition 16.35]) that for $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ and $x\in \mathbb {S}^n$ , the Smale’s projective gamma is given by

$$ \begin{align*}\gamma(f,x):=\sup_{k\geq2} \left\|\frac{1}{k!}\mathrm{D}_xf^\dagger\overline{\mathrm{D}}_x^kf\right\|^{\frac{1}{k-1}},\end{align*} $$

where $\|~\|=\|~\|_{2,2}$ is the operator norm (with respect to Euclidean norms) of a multilinear map.

We now discuss the role of the above properties.

Regularity inequality. The regularity inequality guarantees that when $\mathsf {K}(f,x)<\infty $ , either x is far away from the zero set of f or $\mathrm {D}_xf^\dagger $ exists and is well-defined. The latter is important because it allows us to do various geometric arguments that rely on this pseudoinverse being defined or, equivalently, on $\mathrm {D}_x f$ being surjective. In the particular case of $\mathsf {K}$ , we could state it with equalities (see its proof below), but we leave the statement with inequalities as this is the one holding for $\kappa $ as well and it is enough for our purposes.

1st Lipschitz property. The main use of the 1st Lipschitz inequality is to control the variation of $\mathsf {K}$ with respect to f. This property implies that

(3.7) $$ \begin{align} \frac{1-\frac{\left\|f-\tilde{f}\right\|_\infty^{\mathbb{R}}}{\left\|f\right\|_\infty^{\mathbb{R}}}}{1+\mathsf{K}(f,x)\frac{\left\|f-\tilde{f}\right\|_\infty^{\mathbb{R}}}{\left\|f\right\|_\infty^{\mathbb{R}}}}\mathsf{K}(f,x) \leq\mathsf{K}\left(\tilde{f},x\right) \leq \frac{1+\frac{\left\|f-\tilde{f}\right\|_\infty^{\mathbb{R}}}{\left\|f\right\|_\infty^{\mathbb{R}}}}{1-\mathsf{K}(f,x)\frac{\left\|f-\tilde{f}\right\|_\infty^{\mathbb{R}}}{\left\|f\right\|_\infty^{\mathbb{R}}}}\mathsf{K}(f,x) \end{align} $$

whenever $\mathsf {K}(f,x)\frac {\left \|f-\tilde {f}\right \|_\infty ^{\mathbb {R}}}{\left \|f\right \|_\infty ^{\mathbb {R}}}<1$ . This formula shows how the condition number of an approximation of f relates to that of f.

2nd Lipschitz property. The 2nd Lipschitz property allows us to gauge the variation of $\mathsf {K}$ with respect to x. In this sense, it is very similar to the first Lipschitz property, and it implies that

(3.8) $$ \begin{align} \frac{1}{1+\mathsf{K}(f,x)\mathrm{dist}_{\mathbb{S}}(x,\tilde{x})}\mathsf{K}(f,x) \leq\mathsf{K}\left(f,\tilde{x}\right)\leq \frac{1}{1-\mathsf{K}(f,x)\mathrm{dist}_{\mathbb{S}}(x,\tilde{x})}\mathsf{K}(f,x) \end{align} $$

whenever $\mathsf {K}(f,x)\mathrm {dist}_{\mathbb {S}}(x,\tilde {x})<1$ . Here $\mathrm {dist}_{\mathbb {S}}$ denotes the geodesic distance in $\mathbb {S}^n$ .

Higher derivative estimate. Smale’s projective gamma, $\gamma (f,\zeta )$ , controls many aspects of the local geometry around a zero $\zeta $ of the function f, notably, in the case $q=n$ , the radius of the basin of attraction at $\zeta $ of Newton’s operator $N_f$ associated with f. Recall (see [Reference Bürgisser and Cucker14, Definition 16.34]) that we say $x\in \mathbb {S}^n$ is an approximate zero of $f\in \mathcal {H}_{\boldsymbol {d}}[n]$ with associated zero $\zeta \in \mathbb {S}^n$ when for all $k\geq 1$ , the kth iteration $N_f^k$ of $N_f$ satisfies

$$ \begin{align*}\mathrm{dist}_{\mathbb{S}}(N_f^k,x)\le \left(\frac12\right)^{2^k-1} \mathrm{dist}_{\mathbb{S}}(x,\zeta). \end{align*} $$

We have the following result (see [Reference Bürgisser and Cucker14, Theorem 16.38 and Table 16.1]).

The computation of $\gamma (f,x)$ appears to require all the derivatives of f. The higher derivative estimate allows one to estimate $\gamma (f,x)$ in terms of the first derivative only.

Proof of Theorem 3.2.

Regularity inequality. By definition,

$$ \begin{align*}\frac{1}{\mathsf{K}(f,x)}=\max\left\{\frac{\|f(x)\|}{\sqrt{q}\|f\|_\infty^{\mathbb{R}}},\frac{1}{\sqrt{q}\|f\|_\infty^{\mathbb{R}}\left\|\mathrm{D}_xf^\dagger\Delta\right\|_{2,2}}\right\}.\end{align*} $$

Hence either $\frac {1}{\mathsf {K}(f,x)}=\frac {\|f(x)\|}{\sqrt {q}\|f\|_\infty ^{\mathbb {R}}}$ or $\mathsf {K}(f,x)=\sqrt {q}\|f\|_\infty ^{\mathbb {R}}\left \|\mathrm {D}_xf^\dagger \Delta \right \|_{2,2}$ , which finishes the proof.

1st Lipschitz property. We have that

$$ \begin{align*} \frac{\|g\|_\infty^{\mathbb{R}}}{\mathsf{K}(g,x)} =\max\left\{\frac{\|g(x)\|}{\sqrt{q}}, \frac{\sigma_q\left(\Delta^{-1}\mathrm{D}_xg\right)}{\sqrt{q}}\right\}. \end{align*} $$

Hence, we only need to show that $g\mapsto \|g(x)\|/\sqrt {q}$ and $g\mapsto \sigma _q\left (\Delta ^{-1}\mathrm {D}_xg\right )/\sqrt {q}$ are $1$ -Lipschitz. Now,

$$ \begin{align*} \left|\frac{\|g(x)\|}{\sqrt{q}}-\frac{\left\|\tilde{g}(x)\right\|}{\sqrt{q}}\right|\leq \frac{\left\|\left(g-\tilde{g}\right)(x)\right\|}{\sqrt{q}}\leq \left\|\left(g-\tilde{g}\right)(x)\right\|_{\infty}\leq \left\|g-\tilde{g}\right\|_\infty^{\mathbb{R}}, \end{align*} $$

by the reverse triangle inequality, $\|~\|\leq \sqrt {q}\|~\|_\infty $ and the definition of the real $L_{\infty }$ -norm; and

$$ \begin{align*} \left|\frac{\sigma_q\left(\Delta^{-1}\mathrm{D}_xg\right)}{\sqrt{q}}-\frac{\sigma_q\left(\Delta^{-1}\mathrm{D}_x\tilde{g}\right)}{\sqrt{q}}\right|\leq \frac{\left\|\Delta^{-1}\mathrm{D}_x\left(g-\tilde{g}\right)\right\|_{2,2}}{\sqrt{q}}\leq \left\|\Delta^{-1}\mathrm{D}_x\left(g-\tilde{g}\right)\right\|_{\infty,2}\left\|g-\tilde{g}\right\|_\infty^{\mathbb{R}}, \end{align*} $$

because $\sigma _q$ is $1$ -Lipschitz with respect to $\|~\|_{2,2}$ , $\|~\|\leq \sqrt {q}\|~\|_\infty $ and Kellogg’s inequality (Theorem 2.13). Thus our claims follow.

The claim for $g\mapsto \|g\|_\infty ^{\mathbb {R}}/\mathsf {K}(g)$ follows from the fact that the minimum of a family of $1$ -Lipschitz functions is $1$ -Lipschitz and from

$$ \begin{align*} \frac{\|g\|_\infty^{\mathbb{R}}}{\mathsf{K}(g)} =\min_{x\in\mathbb{S}^n}\frac{\|g\|_\infty^{\mathbb{R}}}{\mathsf{K}(g,x)}. \end{align*} $$

For the lower bound, just note that

$$ \begin{align*}\frac{\|f\|_\infty^{\mathbb{R}}}{\mathsf{K}(f,x)}=\left|\frac{\|f\|_\infty^{\mathbb{R}}}{\mathsf{K}(f,x)}-\frac{\|0\|_\infty^{\mathbb{R}}}{\mathsf{K}(0,x)}\right|\leq \|f-0\|_{\infty}^{\mathbb{R}}=\|f\|_\infty^{\mathbb{R}}\end{align*} $$

by the proven Lipschitz property, so $\mathsf {K}(f,x)\geq 1$ . Similarly with $\mathsf {K}(f)$ .

2nd Lipschitz property. Without loss of generality, assume that $\|f\|_\infty ^{\mathbb {R}}=1$ after scaling f by an appropriate constant; note that this does not change the value of $\mathsf {K}$ . Let $y,\tilde {y}\in \mathbb {S}^n$ and $u\in \mathscr {O}(n+1)$ be the planar rotation taking y into $\tilde {y}$ . Then

$$ \begin{align*} \left|\frac{1}{\mathsf{K}\left(f,y\right)}-\frac{1}{\mathsf{K}\left(f,\tilde{y}\right)}\right|=\left|\frac{1}{\mathsf{K}\left(f,y\right)}-\frac{1}{\mathsf{K}\left(f^u,y\right)}\right|\leq \|f-f^u\|_\infty^{\mathbb{R}}, \end{align*} $$

where $f^u:=f(uX)$ and where the equality follows from the fact that the $L_\infty $ -norm is orthogonally invariant along with the inequality from the 1st Lipschitz property.

Now, arguing as when proving the 1st Lipschitz property, we have that for all $z\in \mathbb {S}^n$ ,

$$ \begin{align*}\left|f(z)-f(uz)\right|\leq \mathbf{D}\,\mathrm{dist}_{\mathbb{S}}(z,uz).\end{align*} $$

By the choice of u, we have that $\mathrm {dist}_{\mathbb {S}}(z,uz)\leq \mathrm {dist}_{\mathbb {S}}(y,\tilde {y})$ . Therefore $\|f-f^u\|_\infty ^{\mathbb {R}}\leq \mathbf {D}\,\mathrm {dist}_{\mathbb {S}}(y,\tilde {y})$ , and we are done.

We note that a variational argument showing that both $y\mapsto \|g(y)\|/\sqrt {q}$ and $y\mapsto \sigma _q(\Delta ^{-1}\mathrm {D}_yf))/\sqrt {q}$ are Lipschitz is possible. This argument would be almost identical to the one used for proving the 1st Lipschitz property but varying the point in the sphere instead of the polynomial. We use the above argument since it is simpler and gives a slightly better bound.

Higher derivative estimate. Again, without loss of generality, we assume that $\|f\|_\infty ^{\mathbb {R}}=1$ , since multiplying f by a scalar affects neither the value of $\mathsf {K}$ nor Smale’s projective gamma. Then

$$ \begin{align*} \left\|\frac{1}{k!}\mathrm{D}_xf^\dagger\overline{\mathrm{D}}_x^kf\right\|&\leq \left\|\mathrm{D}_xf^\dagger\Delta\right\|_{2,2}\left\|\frac{\Delta^{-1}}{k!}\overline{\mathrm{D}}_x^kf\right\|_{2,2} &\text{(inequalities for operator norms)}\\ &\leq \sqrt{q}\left\|\mathrm{D}_xf^\dagger\Delta\right\|_{2,2}\left\|\frac{\Delta^{-1}}{k!}\overline{\mathrm{D}}_x^kf\right\|_{2,\infty} &\|~\|/\sqrt{q}\leq \|~\|_\infty\\ &\leq \mathsf{K}(f,x)\left\|\frac{\Delta^{-1}}{k!}\overline{\mathrm{D}}_x^kf\right\|_{2,\infty} &\text{(assumption + regularity inequality)}\\ &\leq \frac{1}{k}\binom{\mathbf{D}-1}{k-1}\mathsf{K}(f,x).&(\text{Corollary}~{2.20}) \end{align*} $$

Taking $(k-1)$ th roots, we have that $\mathsf {K}(f,x)^{\frac {1}{k-1}}\leq \mathsf {K}(f,x)$ , since $\mathsf {K}(f,x)\geq 1$ by Corollary 2.14, and that

$$ \begin{align*} \left(\frac{1}{k}\binom{\mathbf{D}-1}{k-1}\right)^{\frac{1}{k-1}}\leq \frac{\mathbf{D}-1}{2}, \end{align*} $$

using that $\frac {1}{k}\binom {\mathbf {D}-1}{k-1}\leq (\mathbf {D}-1)^{k-1}/2^{k-1}$ . Putting this together, we obtain the desired bound for Smale’s projective gamma.

The following proposition, which we state here for the sake of completeness, will be proved in Section A.

Proposition 3.4. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $x\in \mathbb {S}^n$ . Then

$$ \begin{align*} \frac{\|f\|_{\infty}^{\mathbb{R}}}{\mathrm{dist}_{\infty}^{\mathbb{R}}(f,\Sigma_{\boldsymbol{d},x}^{\mathbb{R}}[q])}\leq \mathsf{K}(f,x)\leq2\sqrt{\sum_{i=1}^qd_i^2}\,\frac{\|f\|_{\infty}^{\mathbb{R}}}{\mathrm{dist}_{\infty}^{\mathbb{R}}(f,\Sigma_{\boldsymbol{d},x}^{\mathbb{R}}[q])} \end{align*} $$

and

$$ \begin{align*} \frac{\|f\|_{\infty}^{\mathbb{R}}}{\mathrm{dist}_{\infty}^{\mathbb{R}}(f,\Sigma_{\boldsymbol{d}}^{\mathbb{R}}[q])}\leq \mathsf{K}(f)\leq2\sqrt{\sum_{i=1}^qd_i^2}\,\frac{\|f\|_{\infty}^{\mathbb{R}}}{\mathrm{dist}_{\infty}^{\mathbb{R}}(f,\Sigma_{\boldsymbol{d}}^{\mathbb{R}}[q])}, \end{align*} $$

where $\mathrm {dist}_\infty ^{\mathbb {R}}$ is the distance induced by $\|~\|_\infty ^{\mathbb {R}}$ ,

$$ \begin{align*} \Sigma_{\boldsymbol{d},x}^{\mathbb{R}}[q]:=\left\{g\in \mathcal{H}_{\boldsymbol{d}}^{\mathbb{R}}[q]\mid g(x)=0,\,\mathrm{rank}\,\mathrm{D}_x g < q\right\},~\text{ and }~\Sigma_{\boldsymbol{d}}^{\mathbb{R}}[q]:=\bigcup_{x\in\mathbb{S}^n}\Sigma_{\boldsymbol{d},x}^{\mathbb{R}}[q]. \end{align*} $$

3.2 Properties of the complex condition number $\mathsf {M}$

In the complex case, Theorem 3.2 takes the form of the following result, whose proof is identical, so we omit it. We do not consider a regularity inequality for $\mathsf {M}$ since over complex numbers one usually considers $\mathsf {M}(f,\zeta )$ for a zero $\zeta $ of f (or a point nearby).

We finish with the following proposition, which combines the 1st and 2nd Lipschitz properties of $\mathsf {M}$ , as it will play a fundamental role in our analysis of linear homotopy in Section 5. We note that this proposition is to $\mathsf {M}$ what [Reference Bürgisser and Cucker14, Proposition 16.55] is to $\mu _{\mathrm {norm}}$ .

Proposition 3.6. Let $f,\tilde {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ , $\zeta ,\tilde {\zeta }\in \mathbb {P}^n$ and $\varepsilon \in (0,1)$ . If

$$ \begin{align*} \mathsf{M}(f,\zeta)\max\left\{\frac{2\|\tilde{f}-f\|_\infty^{\mathbb{R}}}{\|f\|_{\infty}^{\mathbb{C}}},\mathbf{D}\,\mathrm{dist}_{\mathbb{P}}(\zeta,\tilde{\zeta})\right\}\leq\frac{\varepsilon}{4}, \end{align*} $$

then

$$ \begin{align*}\frac{1}{1+\varepsilon}\mathsf{M}\left(f,\zeta\right)\leq \mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)\leq (1+\varepsilon)\mathsf{M}\left(f,\zeta\right).\end{align*} $$

Proof. Note that

$$ \begin{align*} \left|\frac{1}{\mathsf{M}(f,\zeta)}- \frac{1}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}\right| \leq \left|\frac{1}{\mathsf{M}(f,\zeta)} -\frac{1}{\mathsf{M}\left(\tilde{f},\zeta\right)}\right| +\left|\frac{1}{\mathsf{M}\left(\tilde{f},\zeta\right)} -\frac{1}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}\right|. \end{align*} $$

For the first term in the sum, we have

$$ \begin{align*}\left|\frac{1}{\mathsf{M}\left(f,\zeta\right)} -\frac{1}{\mathsf{M}\left(\tilde{f},\zeta\right)}\right| =\left|\frac{1}{\mathsf{M}\left(\frac{f}{\|f\|_\infty^{\mathbb{C}}},\zeta\right)} -\frac{1}{\mathsf{M}\left(\frac{\tilde{f}} {\|\tilde{f}\|_\infty^{\mathbb{C}}},\zeta\right)}\right| \leq \left\|\frac{f}{\|f\|_\infty^{\mathbb{C}}} -\frac{\tilde{f}}{\|\tilde{f}\|_\infty^{\mathbb{C}}}\right\|_\infty^{\mathbb{C}} \end{align*} $$

by the 1st Lipschitz property of $\mathsf {M}$ (Theorem 3.5). Now,

$$ \begin{align*} \left\|\frac{f}{\|f\|_\infty^{\mathbb{C}}}-\frac{\tilde{f}}{\|\tilde{f}\|_\infty^{\mathbb{C}}}\right\|_\infty^{\mathbb{C}}\leq \left\|\frac{f}{\|f\|_\infty^{\mathbb{C}}}-\frac{\tilde{f}}{\|f\|_\infty^{\mathbb{C}}}\right\|_\infty^{\mathbb{C}}+\left\|\frac{\tilde{f}}{\|f\|_\infty^{\mathbb{C}}}-\frac{\tilde{f}}{\|\tilde{f}\|_\infty^{\mathbb{C}}}\right\|_\infty^{\mathbb{C}}\leq \frac{2\|\tilde{f}-f\|_\infty^{\mathbb{C}}}{\|f\|_\infty^{\mathbb{C}}}. \end{align*} $$

For the second term, we have

$$ \begin{align*} \left|\frac{1}{\mathsf{M}\left(\tilde{f},\zeta\right)}\right| +\left|\frac{1}{\mathsf{M}\left(\tilde{f},\zeta\right)} -\frac{1}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}\right| \leq \mathbf{D}\,\mathrm{dist}_{\mathbb{P}}(\zeta,\tilde{\zeta}) \end{align*} $$

by the 2nd Lipschitz property of $\mathsf {M}$ (Theorem 3.5).

Hence, we have

$$ \begin{align*} \left|\frac{1}{\mathsf{M}(f,\zeta)} -\frac{1}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}\right| \leq \frac{2\|\tilde{f}-f\|_\infty^{\mathbb{C}}}{\|f\|_\infty^{\mathbb{C}}}+ \mathbf{D}\,\mathrm{dist}_{\mathbb{P}}(\zeta,\tilde{\zeta}). \end{align*} $$

By assumption, after multiplying by $\mathsf {M}(f,\zeta )$ , we have

$$ \begin{align*} \left|1-\frac{\mathsf{M}(f,\zeta)}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}\right|\leq \frac{\varepsilon}{2} \end{align*} $$

so, from

$$ \begin{align*} 1-\frac{\mathsf{M}(f,\zeta)}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}\leq \frac{\varepsilon}{2}~\text{ and }~\frac{\mathsf{M}(f,\zeta)}{\mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)}-1\leq \frac{\varepsilon}{2}, \end{align*} $$

we get

$$ \begin{align*} \frac{1}{1+\frac{\varepsilon}{2}}\mathsf{M}(f,\zeta)\leq \mathsf{M}\left(\tilde{f},\tilde{\zeta}\right)\leq \frac{1}{1-\frac{\varepsilon}{2}}\mathsf{M}(f,\zeta). \end{align*} $$

Since $\varepsilon <1$ , the desired inequalities follow.

4.1 A grid-based algorithm and its condition-based complexity

A grid-based algorithm is a subdivision-based method that constructs a grid to discretise the original problem and solves the latter by working on the grid points only (selecting and finding proximity relations between its points). The algorithms in [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick, Malajovich and Wschebor26, Reference Cucker, Krick, Malajovich and Wschebor27], [Reference Cucker, Krick and Shub28] and [Reference Bürgisser, Cucker and Lairez15, Reference Bürgisser, Cucker and Tonelli-Cueto16, Reference Bürgisser, Cucker and Tonelli-Cueto17] (compare to [Reference Tonelli-Cueto55]) are grid-based. Their basic structure is (simplifying to the extreme) the following:

1. 1. Estimate the condition number of the problem (with a sequence of grids of increasing fineness).

2. 2. Create an extra grid (if necessary) whose mesh is determined by the condition number.

3. 3. Select points in the grid, and use them to obtain a solution to the problem.

In general, grid-based algorithms have complexity $\Omega (\mathbf {D}^n)$ . This fact allows us to estimate the norm $\|f\|_\infty ^{\mathbb {R}}$ of the data f without affecting the overall complexity of the algorithms. Moreover, the fact that $\mathsf {K}$ is smaller than $\kappa $ results in a cost reduction.

In this subsection, we focus on an algorithm for the computation of the Betti numbers of a spherical algebraic set. This covers the case of counting zeros of a square polynomial system treated in [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick, Malajovich and Wschebor26, Reference Cucker, Krick, Malajovich and Wschebor27] and the computation of the Betti numbers of a projective real variety [Reference Cucker, Krick and Shub28]. For simplicity of exposition, we omit some computational aspects: 1) our presentation of the algorithms follows the construction-selection paradigm of [Reference Bürgisser, Cucker and Lairez15, Reference Bürgisser, Cucker and Tonelli-Cueto16, Reference Bürgisser, Cucker and Tonelli-Cueto17] instead of the inclusion-exclusion paradigm of [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick, Malajovich and Wschebor26, Reference Cucker, Krick, Malajovich and Wschebor27, Reference Cucker, Krick and Shub28]. This makes the exposition of the algorithms easier without compromising their computational complexity. 2) We focus on Betti numbers to avoid describing the more involved computation of torsion coefficients in the homology groups. 3) We deal with neither parallelisation nor finite precision. The interested reader can find details about these in the cited references.

The backbone of existing grid-based algorithms in numerical real algebraic geometry [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick, Malajovich and Wschebor26, Reference Cucker, Krick, Malajovich and Wschebor27, Reference Cucker, Krick and Shub28, Reference Bürgisser, Cucker and Lairez15, Reference Bürgisser, Cucker and Tonelli-Cueto16, Reference Bürgisser, Cucker and Tonelli-Cueto17] is an effective construction of spherical nets. The basic construction was done originally in [Reference Cucker, Krick, Malajovich and Wschebor25] and is based on projecting the uniform grid in the boundary of a unit cube onto the unit sphere.

Recall that a (spherical) $\delta $ -net is a finite subset $\mathcal {G}\subset \mathbb {S}^n$ such that for all $x\in \mathbb {S}^n$ , $\mathrm {dist}_{\mathbb {S}}(x,\mathcal {G})< \delta $ . We will omit the term ‘spherical’ as all nets we consider are so.

Proposition 4.1. There is an algorithm GRID that on input $(n,k)\in \mathbb {N}\times \mathbb {N}$ , outputs a $2^{-k}$ -net $\mathcal {G}_k\subset \mathbb {S}^n$ with

$$ \begin{align*} |\mathcal{G}_k|=\mathcal{O}\left(2^{n\log n+nk}\right). \end{align*} $$

The cost of this algorithm is $\mathcal {O}\left (2^{n\log n+nk}\right )$ .

4.1.1 Computation of $\Vert~\Vert_\infty ^{\mathbb {R}}$

The following is an easy consequence of Kellogg’s theorem.

Proposition 4.3. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $\mathcal {G}\subset \mathbb {S}^n$ be a $\delta $ -net. If $\mathbf {D} \delta <\sqrt {2}$ , then

$$ \begin{align*} \max_{x\in\mathcal{G}}\|f(x)\|_\infty\leq \|f\|_\infty^{\mathbb{R}}\leq \frac{1}{1-\frac{\mathbf{D}^2}{2}\delta^2} \max_{x\in\mathcal{G}}\|f(x)\|_\infty. \end{align*} $$

Proof. We only need to show the right-hand inequality, the other being trivial. Without loss of generality, assume that $q=1$ : that is, f is a homogeneous polynomial of degree $\mathbf {D}$ .

Let $x_*$ be the maximum of $|f|$ on $\mathbb {S}^n$ , $x\in \mathcal {G}$ such that $\mathrm {dist}_{\mathbb {S}}(x_\ast ,x)\leq \delta $ and $[0,1]\ni t\mapsto x_t$ the geodesic on $\mathbb {S}^n$ going from $x_*$ to x with constant speed. Then for the function $t\mapsto M(t):=f(x_t)$ , we have that $|M(1)|\leq |M(0)|+| M'(0)|+\max _{s\in [0,1]}\frac {M"(s)}{2}$ by Taylor’s theorem. Furthermore, $|M(0)|=|f(x_*)|=\|f\|_\infty ^{\mathbb {R}}$ , $|M(1)|=|f(x)|$ and $M'(0)=0$ . The latter is because $x_*$ is an extremal point of f and so of M. Now

$$ \begin{align*}M"(t)=\overline{\mathrm{D}}_{x_t}^2f(\dot x_t,\dot x_t)-\mathbf{D} f(x_t)\mathrm{dist}_{\mathbb{S}}(x_\ast,x)^2,\end{align*} $$

since $\ddot x_t=-\mathrm {dist}_{\mathbb {S}}(x_\ast ,x)^2x_t$ , as $x_t$ is a geodesic on $\mathbb {S}^n$ of constant speed $\mathrm {dist}_{\mathbb {S}}(x_\ast ,x)$ and $\overline {\mathrm {D}}_{x_t}f(x_t)=\mathbf {D} f(x_t)$ by Euler’s formula in equation (2.4). Then by Corollary 2.20,

$$ \begin{align*} \max_{s\in [0,1]}\frac{|M"(s)|}{2}\leq \binom{\mathbf{D}}{2}\|f\|_\infty^{\mathbb{R}}+\frac{\mathbf{D}}{2}\|f\|_\infty^{\mathbb{R}} =\frac{\mathbf{D}^2}{2}\|f\|_\infty^{\mathbb{R}}. \end{align*} $$

Thus $\|f\|_\infty ^{\mathbb {R}}\leq |f(x)|+\frac {\mathbf {D}^2}{2}\|f\|_\infty ^{\mathbb {R}}\delta ^2$ , and the desired inequality follows.

Remark 4.4. Proposition 4.3 is a slight improvement of [Reference Ergür, Paouris and Rojas33, Lemma 2.5].

Proposition 4.3 suggests the following algorithm.

Proposition 4.5. Algorithm is correct. On input $(f,k)\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]\times \mathbb {N}$ , its cost is bounded by

$$ \begin{align*} \mathcal{O}\left(2^{n\log n}\mathbf{D}^n2^{\frac{(k+1)n}{2}}N\right). \end{align*} $$

Proof. This is a direct consequence of Propositions 4.1 and 4.3 and the fact that f can be evaluated at $x\in \mathbb {S}^n$ with $\mathcal {O}(N)$ arithmetic operations (see [Reference Bürgisser and Cucker14, Lemma 16.31]).

Remark 4.6. The ideas here can also be applied to compute $\|f\|_\infty ^{\mathbb {C}}$ .

4.1.2 Estimation of $\mathsf {K}$

In many grid-based algorithms, the estimation of condition numbers is done implicitly along the way; this does not affect the overall computational cost, and it makes for an easier understanding of these algorithms. The next proposition is the core of the estimation of $\mathsf {K}$ . Note that the mesh of the grid needed to estimate $\mathsf {K}$ depends on $\mathsf {K}$ itself.

Proposition 4.7. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $\mathcal {G}\subset \mathbb {S}^n$ be a $\delta $ -net. If

$$ \begin{align*}\delta\,\mathbf{D}\max_{x\in\mathcal{G}}\mathsf{K}(f,x)<1,\end{align*} $$

then

$$ \begin{align*} \max_{x\in\mathcal{G}}\mathsf{K}(f,x)\leq \mathsf{K}(f)\leq \frac{1}{1-\delta\,\mathbf{D}\,\max_{x\in\mathcal{G}}\mathsf{K}(f,x)} \max_{x\in\mathcal{G}}\mathsf{K}(f,x). \end{align*} $$

Proof. We only have to prove the right-hand side inequality since the other one is obvious. Let $x_\ast \in \mathbb {S}^n$ such that $\mathsf {K}(f)=\mathsf {K}(f,x_\ast )$ and $x\in \mathcal {G}$ such that $\mathrm {dist}_{\mathbb {S}}(f,x)\leq \delta $ . Then by the 2nd Lipschitz property (Theorem 3.2), we have

$$ \begin{align*} \frac{1}{\mathsf{K}(f,x)}-\frac{1}{\mathsf{K}(f,x_\ast)}\leq \mathbf{D}\,\mathrm{dist}_{\mathbb{S}}(x_\ast,x)\leq \mathbf{D}\,\delta. \end{align*} $$

Hence $1/\mathsf {K}(f,x_\ast )\leq (1-\delta \,\mathbf {D}\,\mathsf {K}(f,x))/\mathsf {K}(f,x)$ , and the desired inequality follows from the hypothesis.

Proposition 4.7 suggests the following algorithm, which involves only one $L_\infty $ -norm computation.

Proposition 4.8. Algorithm is correct. On input $(f,k,b)\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]\times \mathbb {N}\times (\mathbb {N}\cup \{\infty \})$ , its cost is bounded by

$$ \begin{align*} 2^{\mathcal{O}(n(k+\log n))}D^n N\min\{\mathsf{K}(f)^n,2^{nb}\}. \end{align*} $$

Proof. The correctness follows from Propositions 4.5 and 4.7 and $(1-2^{-(k+1)})^2>1-2^{-k}$ .

The cost of the first line of the algorithm is bounded by Proposition 4.5. The number of evaluations of

$$ \begin{align*}\sqrt{q}t/\max\{\|f(x)\|,\|\mathrm{D}_x f^\dagger \Delta\|^{-1}\} \end{align*} $$

in the $\ell $ th iteration of the loop is given by Proposition 4.1. We need $\mathcal {O}(N+n^3)$ operations for each such evaluation, by [Reference Bürgisser and Cucker14, Proposition 16.32].

In this way, if the loop runs $\ell _0$ iterations, it performs a total of

$$ \begin{align*} \mathcal{O}(2^{n\log n}(D^n2^{\frac{(k+2)n}{2}}N+2^{n(\ell_0+1)}(N+n^3))) \end{align*} $$

operations.

If the algorithm outputs $\mathcal {K}$ , then $\ell _0=\lceil k+\log \mathbf {D}+ \log \mathcal {K}-\log (1-2^{-k})\rceil $ . Moreover, from the correctness, $\log \mathcal {K}-\log (1-2^{-k})\leq \log \mathsf {K}(f)$ , so $\ell _0\leq k+1+\log \mathbf {D}+\log \mathsf {K}(f)$ .

If the algorithm outputs fail, then the first criterion had to fail, so as long as the second criterion fails too, we have

$$ \begin{align*} \ell<k+\log \mathbf{D}+b. \end{align*} $$

So, in this case, $\ell _0\leq k+1+\log \mathbf {D}+\log b$ .

We conclude from the bounds above and some straightforward computations.

By setting k to $7$ and $b=\infty $ , we have the following important corollary.

Corollary 4.9. There is an algorithm $\mathsf {K}$ -Estimate $^*$ that on input $(f)\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ computes $\mathcal {K}\in [1,\infty )$ such that

$$ \begin{align*}0.99\mathcal{K}\leq\mathsf{K}(f)\leq\mathcal{K}.\end{align*} $$

This algorithm halts if and only if $\mathsf {K}(f)<\infty $ , and its cost is bounded by

(□) $$ \begin{align*} 2^{\mathcal{O}(n\log n)}D^n N\mathsf{K}(f)^n. \end{align*} $$

4.1.3 Complexity analysis of grid-based algorithms using $\mathsf {K}$

To get the grid method to work, we need two ingredients: a method for selecting the points in the grid near the geometric object of interest and a way of controlling distances between these two sets.

Theorem 4.10 (Construction-selection).

Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $\mathcal {G}\subseteq \mathbb {S}^n$ be a $\delta $ -net. If

$$ \begin{align*} 4\mathbf{D}^2\mathsf{K}(f)^2\delta<1 \end{align*} $$

and $Q\in \mathbb {R}$ is such that $0.99Q\leq \|f\|_\infty ^{\mathbb {R}}\leq Q$ , then

$$ \begin{align*} \mathrm{dist}_H\left(\left\{x\in \mathcal{G}\mid \frac{\|f(x)\|}{\sqrt{q}Q} < \mathbf{D}\,\delta\right\},\mathcal{Z}_{\mathbb{S}}(f)\right)< 2\mathbf{D}\mathsf{K}(f)\delta, \end{align*} $$

where $\mathrm {dist}_H(A,B):=\max \{\sup \{\mathrm {dist}(a,B)\mid a\in A\}, \sup \{\mathrm {dist}(b,A)\mid b\in B\}\}$ is the Hausdorff distance.

Following [Reference Federer35], recall that the medial axis $\Delta _X$ of a closed set $X\subset \mathbb {R}^n$ is the set

$$ \begin{align*} \Delta_X:=\{p\in\mathbb{R}^n\mid \#\{x\in X\mid \mathrm{dist}(p,x)=\mathrm{dist}(p,X)\}\geq 2\} \end{align*} $$

consisting of those points for which there is more than one nearest point in X and that the reach $\tau (X)$ of X is the quantity

$$ \begin{align*} \tau(X):=\mathrm{dist}(X,\Delta_X) \end{align*} $$

measuring the size of the neighbourhood of X within which the nearest point projection is well-defined. If X is finite, then $\Delta _X$ is the union of the boundaries of the cells of the Voronoi diagram of X, and $\tau (X)$ is half the minimum distance between two distinct points of X. Thus, when $\mathcal {Z}_{\mathbb {S}}(f)$ is zero-dimensional, $2\tau (\mathcal {Z}_{\mathbb {S}}(f))$ is the separation of the zeros of f in the sphere.

Theorem 4.11. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ . Then

$$ \begin{align*}\tau(\mathcal{Z}_{\mathbb{S}}(f))\geq \frac{1}{7\mathbf{D}\mathsf{K}(f)}.\end{align*} $$

Proof of Theorem 4.10.

Let $x_0\in \mathcal {Z}_{\mathbb {S}}(f)$ . Then there is some $x_1\in \mathcal {G}$ such that $\mathrm {dist}_{\mathbb {S}}(x_0,x_1)\leq \delta $ . Let $[0,1]\ni t\mapsto x_t$ be the geodesic joining them. By Taylor’s theorem,

$$ \begin{align*}\|f(x_1)\|\leq \|f(x_0)\|+ \delta \sup_{s\in [0,1]}\|\mathrm{D}_xf\|,\end{align*} $$

so, by Kellogg’s theorem (Corollary 2.14) and $f(x_0)=0$ , we have that $\frac {\|f(x_1)\|}{\sqrt {q}\|f\|_\infty ^{\mathbb {R}}}\leq \mathbf {D}\,\delta $ . Hence $\frac {\|f(x_1)\|}{\sqrt {q}Q}\leq \mathbf {D}\,\delta $ and

$$ \begin{align*} \mathrm{dist}\left(x_0,\left\{x\in \mathcal{G}\mid \frac{\|f(x)\|}{\sqrt{q}Q}\leq \mathbf{D}\,\delta\right\}\right)\leq \mathrm{dist}(x_0,x_1)\le \mathrm{dist}_{\mathbb{S}}(x_0,x_1)\leq\delta. \end{align*} $$

Now let $x_2\in \mathcal {G}$ be such that $\frac {\|f(x_2)\|}{\sqrt {q}Q}< \mathbf {D}\,\delta $ . Then

(4.1) $$ \begin{align} \frac{\|f(x_2)\|}{\sqrt{q}\|f\|_\infty^{\mathbb{R}}} < 1.02\mathbf{D}\,\delta \le \frac{1}{4\mathbf{D}\mathsf{K}(f)^2} <\frac{1}{\mathsf{K}(f,x_2)}, \end{align} $$

the second inequality by our hypothesis. Because of the regularity inequality (Theorem 3.2), we must then have $\sqrt {q}\|f\|^{\mathbb {R}}_{\infty }\|\mathrm {D}_{x_2}f^{\dagger }\Delta ^{1/2}\| \le \mathsf {K}(f,x_2)$ . It follows that

$$ \begin{align*} \|\mathrm{D}_{x_2}f^{\dagger}f(x_2)\|\gamma(f,x_2) &< \frac{\mathsf{K}(f,x_2)}{\sqrt{q}\|f\|^{\mathbb{R}}_{\infty}}\gamma(f,x_2) \le \frac{1}{2}\mathbf{D}\mathsf{K}(f)^2\frac{\|f(x_2)\|}{\sqrt{q}\|f\|_\infty^{\mathbb{R}}}\\ &< \frac{1.02}{2}\mathbf{D}^2\mathsf{K}(f)^2\delta< \frac{1.02}{8} < 0.13071\ldots \end{align*} $$

where we used the higher derivative estimate (Theorem 3.2) in the first line and equation (4.1) and the hypothesis in the second. This means Smale’s $\alpha $ -criterion holds for $x_2$ and $f_{|\mathrm {T}_{x_0}\mathbb {S}^n}$ by [Reference Dedieu29, Théorème 128]. Hence there is $x_3\in \mathrm {T}_{x_2}\mathbb {S}^n$ such that $f(x_3)=0$ and

$$ \begin{align*} \mathrm{dist}(x_2,x_3)\leq 1.64 \|\mathrm{D}_{x_2}f^{\dagger}f(x_2)\| \leq 1.64\cdot 1.02\,\mathbf{D}\mathsf{K}(f)\delta < 2\mathbf{D}\mathsf{K}(f)\delta. \end{align*} $$

Since $\mathrm {dist}(x_2,x_3/\|x_3\|) =\arctan \mathrm {dist}(x_2,x_3)\leq \mathrm {dist}(x_2,x_3)$ , we are done.

Remark 4.12. The proof also shows the convergence of Newton’s method associated with $f_{|\mathrm {T}_x\mathbb {S}^n}$ for every $x\in \mathcal {G}$ such that $\frac {\|f(x)\|}{\sqrt {q}\|f\|_\infty ^{\mathbb {R}}}\leq \mathbf {D}\,\delta $ . Hence, we can refine our approximations if needed.

Sketch of proof of Theorem 4.11.

The proof is very similar to the one of [Reference Bürgisser, Cucker and Lairez15, Theorem 4.12]. By [Reference Bürgisser, Cucker and Lairez15, Lemma 2.7] and [Reference Bürgisser, Cucker and Lairez15, Theorem 3.3], we have that

$$ \begin{align*} \tau(\mathcal{Z}_{\mathbb{S}}(f))\geq \min\left\{1,\frac{1}{14\max\{\gamma(f,x)\mid x\in \mathcal{Z}_{\mathbb{S}}(f)\}}\right\}. \end{align*} $$

Hence, by the higher derivative estimate (Theorem 3.2), the desired bound follows. $\Box $

The following theorem is a variant of the so-called Niyogi-Smale-Weinberger theorem [Reference Niyogi, Smale and Weinberger47, Proposition 7.1].

Theorem 4.13. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ , $\mathcal {G}\subset \mathbb {S}^n$ be a $\delta $ -net and $Q\in \mathbb {R}$ be such that $0.99Q\leq \|f\|_\infty ^{\mathbb {R}}\leq Q$ . If $90\mathbf {D}^2\mathsf {K}(f)^2\delta <1$ , then for every

$$ \begin{align*} \varepsilon\in \left(6\mathbf{D}\mathsf{K}(f)\delta,\frac{1}{14\mathbf{D}\mathsf{K}(f)}\right), \end{align*} $$

the sets $\mathcal {Z}_{\mathbb {S}}(f)$ and

$$ \begin{align*} \bigcup\left\{B(x,\varepsilon)\mid x\in\mathcal{G},\,\frac{\|f(x)\|}{\sqrt{q}Q}<\mathbf{D}\delta\right\} \end{align*} $$

are homotopically equivalent. In particular, they have the same Betti numbers.

Proof. This is just [Reference Bürgisser, Cucker and Lairez15, Theorem 2.8] combined with Theorems 4.10 and 4.11.

We can now describe the algorithm. We will call a black box Betti for computing the Betti numbers of a union of balls. This is a standard procedure in topological data analysis [Reference Edelsbrunner32].

Proposition 4.14. Algorithm is correct, and its cost is bounded by

$$ \begin{align*} 2^{\mathcal{O}(n^2\log n)}D^{10n^2}\mathsf{K}(f)^{10n^2}. \end{align*} $$

Proof. Correctness is a consequence of Theorem 4.13 and the fact that the computed Q satisfies $0.99Q\leq \|f\|_\infty ^{\mathbb {R}}\leq Q$ by Proposition 4.5.

For the complexity, we apply Proposition 4.3 for the first line, Corollary 4.9 for the second line and Proposition 4.1 for the fourth and fifth lines. We know that has cost $\mathcal {O}\left (2^{\mathcal {O}(n\log n)}|\mathcal {X}|^{5n}\right )$ (see [Reference Cucker, Krick and Shub28, Section 5] for example) and that $|\mathcal {X}|=\mathcal {O}(2^{n\log n}\mathbf {D}^{2n}\mathsf {K}(f)^{2n})$ , by Proposition 4.1. Note that we have eliminated N from the bounds. We have done so using the fact that as $q\le n$ (by the precondition of the input), $N\leq 2^{n\log n}\mathbf {D}^{n}$ .

We note that our bound uses $\mathcal {K}\leq 1.02\mathsf {K}(f)$ to get the cost dependent on $\mathsf {K}(f)$ instead of on the computed estimate $\mathcal {K}$ .

The complexity estimate in Proposition 4.14 does not differ much from those in other grid-based algorithms. We will see in Section 4.3, however, that the occurrence of $\mathsf {K}$ in the place of $\kappa $ leads to substantial improvements when one goes beyond the worst-case framework and considers random input models.

4.2 Complexity of the Plantinga-Vegter algorithm

The ideas above can also be applied to the Plantinga-Vegter algorithm [Reference Plantinga and Vegter49]. In a recent work [Reference Cucker, Ergür and Tonelli-Cueto24] (compare to [Reference Cucker, Ergür and Tonelli-Cueto23]), we performed an extensive analysis of this algorithm, including details for finite precision arithmetic. So we will be brief here, referring the reader to [Reference Cucker, Ergür and Tonelli-Cueto24] for details, and will only focus on the (exact) interval version of the algorithm.

4.2.1 The Plantinga-Vegter subdivision algorithm

Let $\mathcal {P}_{d}$ be the space of polynomials in $X_1,\ldots ,X_n$ of degree at most d. The Plantinga-Vegter algorithm [Reference Plantinga and Vegter49]Footnote ² is a subdivision-based algorithm for obtaining a piecewise linear approximation of the zero set of $f\in \mathcal {P}_{d}$ inside $[-a,a]^n$ . As customary, we will focus on the complexity analysis of the subdivision routine only. The idea is to iteratively subdivide some boxes – that is, sets of the form $B=m(B)+[-w(B)/2,w(B)/2]^n$ (here $m(B)\in \mathbb {R}^n$ is the centre of B and $w(B)>0$ is its width) – in $[-a,a]^n$ until every box B in the subdivision satisfies the following condition:

$$ \begin{align*} C_f(B)\,:\,\text{either }0\notin f(B)\text{ or }0\notin \langle \nabla f(B),\nabla f(B)\rangle, \end{align*} $$

where $\langle ~,~\rangle $ is the standard inner product and $\nabla f$ is the gradient vector of f. Once this criterion is satisfied by all boxes in the subdivision, the Plantinga-Vegter algorithm returns a topologically accurate approximation of the zero set of f in the region $[a,-a]^n$ and halts (see [Reference Plantinga and Vegter49] ( $n\leq 3$ ) and [Reference Galehouse37] (arbitrary n) for details on how this is done).

For $f\in \mathcal {P}_{d}$ , we define

$$ \begin{align*} \|f\|_\infty:=\max\{|f^{\mathsf{h}}(x)|\mid x\in\mathbb{S}^n\}=\|f^{\mathsf{h}}\|_\infty^{\mathbb{R}}, \end{align*} $$

where $f^{\mathsf {h}}\in \mathcal {H}_d[1]$ is the homogenisation of f. Taking the maps (2.3), (2.4), (2.5) in [Reference Cucker, Ergür and Tonelli-Cueto24] and substituting on them the Weyl norm by the real $L_\infty $ -norm, we get

(4.2) $$ \begin{align} h(x)=\frac{1}{\|f\|_\infty(1+\|x\|^2)^{(d-1)/2}} \quad\text{ and }\quad h'(x)=\frac{1}{d\|f\|_\infty(1+\|x\|^2)^{d/2-1}} \end{align} $$

together with

(4.3) $$ \begin{align} \widehat{f}:x\mapsto h(x)f(x)=\frac{f(x)}{\|f\|_\infty(1+\|x\|^2)^{(d-1)/2}} \end{align} $$

and

(4.4) $$ \begin{align} \widehat{\nabla f}:x\mapsto h'(x)\mathrm{D} f(x) =\frac{\nabla f(x)}{d\|f\|_\infty(1+\|x\|^2)^{d/2-1}}. \end{align} $$

One can use these maps to produce interval approximations as we do in [Reference Cucker, Ergür and Tonelli-Cueto24]. For $X\subseteq \mathbb {R}^m$ , we denote by $\square X$ the set of boxes contained in X. Recall that an interval approximation of $f:\mathbb {R}^n\rightarrow \mathbb {R}^q$ is a function $\square f:\square \mathbb {R}^n\rightarrow \square \mathbb {R}^q$ that maps boxes in $\mathbb {R}^n$ to boxes in $\mathbb {R}^q$ in such a way that $f(B)\subseteq \square f(B)$ .

Proposition 4.15. Let $f\in \mathcal {P}_{d}$ . Then

$$ \begin{align*}\square[hf]:B\mapsto \widehat{f}(m(B))+(1+d)\sqrt{n}\left[-\frac{w(B)}{2},\frac{w(B)}{2}\right] \end{align*} $$

is an interval approximation of $hf$ and

$$ \begin{align*}\square[\|h'\mathrm{D} f\|]:B\mapsto \|\widehat{\nabla f}(m(B))\|+d\sqrt{n}\left[-\frac{w(B)}{2},\frac{w(B)}{2}\right]\end{align*} $$

is an interval approximation of $\|h'\mathrm {D} f\|$ .

Sketch of proof.

Using the bounds from Kellogg’s theorem (Theorem 2.13) and its corollaries, we can easily deduce (as is done in the proof of Theorem 3.2) that the maps

$$ \begin{align*} g/\|g\|_\infty^{\mathbb{R}}:\mathbb{S}^n\rightarrow [-1,1]~\text{ and } \overline{\mathrm{D}} g(v)/(d\|g\|_\infty^{\mathbb{R}}\|v\|):\mathbb{S}^n\rightarrow [-1,1] \end{align*} $$

are d- and $(d-1)$ -Lipschitz (with respect to the geodesic distance) for $g\in \mathcal {H}_d^{\mathbb {R}}[1]$ . $\Box $

We now argue as in [Reference Cucker, Ergür and Tonelli-Cueto24, Section 4], but using these Lipschitz properties, to prove that $\hat f$ and $\widehat {\nabla f}$ are $(1+d)$ - and d-Lipschitz, respectively. For the latter, we use the fact that for $v\in \mathbb {R}^n$ , $\overline {\mathrm {D}}_Xf^{\mathsf {h}}\begin {pmatrix}0\\v\end {pmatrix} =(\langle \nabla f,v\rangle )^{\mathsf {h}}$ and that $\|\widehat {\nabla f}\|$ is d-Lipschitz if $\langle \widehat {\nabla f},v\rangle $ is so for every $v\in \mathbb {S}^{n-1}$ .

Using the interval approximations and their Lipschitz properties in Proposition 4.15, we can rewrite the condition $C_f(B)$ . We only need to use [Reference Cucker, Ergür and Tonelli-Cueto24, Lemma 4.2] for the second clause of the condition.

Theorem 4.16. Let $B\in \square \mathbb {R}^n$ . If the condition

$$ \begin{align*} C_f^{\square}(B)\,:\, \left|\widehat{f}(m(B))\right|>2d\sqrt{n}w(B) \text{ or }~\left\|\widehat{\nabla f} (m(B))\right\|>2\sqrt{2}d\sqrt{n}w(B) \end{align*} $$

is satisfied, then $C_f(B)$ is true.

The subdivision procedure of the Plantinga-Vegter algorithm thus takes the following form, where StandardSubdivision is a procedure that, given a box, divides it into $2^n$ equal boxes. Recall that $\square [-a,a]^n$ is the set of boxes within $[-a,a]^n$ .

4.2.2 Complexity of PV-Interval ${}_\infty $

Without much effort, [Reference Cucker, Ergür and Tonelli-Cueto24, Proposition 5.1] transforms into the following proposition. The essential step is multiplying the inequalities in that proposition by $\|f^{\mathsf {h}}\|_W/\|f\|_\infty $ .

Proposition 4.17. Let $f\in \mathcal {P}_{d}$ and $x\in \mathbb {R}^n$ . Then either

where

.

With Proposition 4.17 and the Lipschitz properties shown for $\hat f$ and $\widehat {\nabla f}$ , one can produce a local size bound for $C^{\Box }_f(B)$ . This is a function that, evaluated at a point x, gives a lower bound on the volume of any possible box containing x and not satisfying the predicate $C^{\prime }_f(B)$ .

Theorem 4.18. The map

is a local size bound for $C_f^{\Box }$ (of Theorem 4.16).

Then using the continuous amortisation of [Reference Burr, Krahmer and Yap20, Reference Burr18, Reference Burr, Gao and Tsigaridas19] (see [Reference Cucker, Ergür and Tonelli-Cueto24, Theorem 6.1]), we conclude the following, which takes into account the cost of calling NormApprox $\mathbb {R}$ (Proposition 4.3).

Theorem 4.19. The number of boxes in the final subdivision $\mathcal {S}$ of PV-Interval ${}_\infty $ on input $(f,a)$ is at most

The number of arithmetic operations performed by PV-Interval ${}_\infty $ on input $(f,a)$ is at most

The condition-based estimates in Theorem 4.19 are very similar to those of [Reference Cucker, Ergür and Tonelli-Cueto24, Theorem 6.3]. It is important to observe that only one norm computation is performed by PV-Interval ${}_\infty $ (in its very first step) and that the cost of this computation is already included in the cost bound in Theorem 4.19. We will see in Section 4.3.3 that the occurrence of $\mathsf {K}$ in the place of $\kappa $ results in significant improvements in overall complexity when we consider average or smoothed analysis.

4.3 Probabilistic analysis of algorithms

In the preceding sections, we have shown that existing grid-based and subdivision-based algorithms that use (in their design and/or analysis) $\kappa $ can be modified to use $\mathsf {K}$ instead. Moreover, we have shown that the condition-based complexity estimates in terms of $\mathsf {K}$ are similar to those in terms of $\kappa $ . In this section, we will show that when we consider random inputs, in contrast, the cost (expected or in probability) substantially decreases.

We first introduce the randomness model along with some useful probabilistic results. Then we prove a general comparison result showing that when substituting $\kappa $ by $\mathsf {K}$ , one can expect to reduce the size of the condition number by a factor of $\sqrt {N}$ . Finally, we apply these estimates to both PolyBetti and the Plantinga-Vegter algorithm and highlight the complexity improvements.

For most algorithms in real algebraic geometry, condition-based estimates show a dependence on either $\kappa ^n$ or $\mathsf {K}^n$ . When this occurs, the complexity estimates improve by a factor of the form $N^{\frac {n}{2}}$ when we pass from $\kappa $ to $\mathsf {K}$ . The final complexity estimates thus change from having an exponent quadratic in n to an exponent quasilinear in n.

4.3.1 The randomness model: dobro random polynomials

Given a random variable $\mathfrak {x}\in \mathbb {R}$ , we say that:

1. (i) $\mathfrak {x}$ is centred if $\mathop {\mathbb {E}}\mathfrak {x}=0$ .

2. (ii) $\mathfrak {x}$ is subgaussian if there is a constant $K>0$ such that for all $p\geq 1$ ,

   $$ \begin{align*} \left(\mathop{\mathbb{E}}|\mathfrak{x}|^p\right)^{\frac{1}{p}}\leq K\sqrt{p}. \end{align*} $$

   The smallest K satisfying this condition is called the $\psi _2$ -norm of $\mathfrak {x}$ and is denoted $\|\mathfrak {x}\|_{\psi _2}$ .

3. (iii) $\mathfrak {x}$ has the anti-concentration property with constant $\rho $ if for all $u\in \mathbb {R}$ and $\varepsilon>0$ ,

   $$ \begin{align*}\mathbb{P}(|\mathfrak{x}-u|<\varepsilon)\leq 2\rho\varepsilon.\end{align*} $$

   Note that this is equivalent to $\mathfrak {x}$ having a density (with respect to the Lebesgue measure) bounded by $\rho $ .

We now extend to tuples the class of real random polynomials introduced in [Reference Cucker, Ergür and Tonelli-Cueto23].

Definition 4.20. A dobro random polynomial tuple $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ with parameters K and $\rho $ is a tuple of random polynomials

$$ \begin{align*} \left(\sum_{|\alpha|=d_1}\binom{d_1}{\alpha}^{\frac{1}{2}} \mathfrak{c}_{1,\alpha}X^\alpha,\ldots,\sum_{|\alpha|=d_q} \binom{d_q}{\alpha}^{\frac{1}{2}}\mathfrak{c}_{q,\alpha}X^\alpha\right) \end{align*} $$

such that the $\mathfrak {c}_{i,\alpha }$ are independent centred subgaussian random variables with $\psi _2$ -norm at most K and anti-concentration property with constant $\rho $ .

Remark 4.21. Probabilistic estimates for a dobro polynomial $\mathfrak {f}$ will depend on $K\rho $ . This product is invariant under scalar multiplication of $\mathfrak {f}$ since $\lambda \mathfrak {f}$ is dobro with parameters $|\lambda |K$ and $\rho /|\lambda |$ . Moreover, note thatFootnote ³ $6K\rho \geq 1$ .

Example 4.22. A dobro random polynomial tuple $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ such that the $\mathfrak {c}_\alpha $ are are independent and identically distributed normal random variables of mean zero and variance one is called a KSS (real) polynomial tuple.Footnote ⁴ In this case, we can take $K\rho =2/\sqrt {\pi }$ .

Example 4.23. A dobro random polynomial tuple $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ such that the $\mathfrak {c}_\alpha $ are are independent and identically distributed uniform random variables in $[-1,1]$ is a Weyl uniform (real) polynomial tuple. In this case, we can take $K\rho =1/2$ .

We now state and prove several probabilistic results that will be used later.

Proposition 4.24 (Subgaussian tail bounds).

Let $\mathfrak {x}\in \mathbb {R}$ be a random variable.

1. 1. If $\mathfrak {x}$ is subgaussian with $\psi _2$ -norm at most K, then for all $t>0$ , $\mathbb {P}(|\mathfrak {x}|\geq t)\leq \mathrm {e}^{1-\frac {t^2}{6K^2}}$ .

2. 2. If there are $C\geq \mathrm {e}$ and $K>0$ such that for all $t>0$ , $\mathbb {P}(|\mathfrak {x}|\geq t)\leq C\mathrm {e}^{-\frac {t^2}{K^2}}$ , then $\mathfrak {x}$ is subgaussian with $\psi _2$ -norm at most $K\left (\sqrt {\pi /2}+\sqrt {2\ln C}\right )$ .

Proposition 4.25 (Hoeffding inequality).

Let $\mathfrak {x}\in \mathbb {R}^N$ be a random vector such that its components $\mathfrak {x}_i$ are centred subgaussian random variables with $\psi _2$ -norm at most K and $a\in \mathbb {S}^{N-1}$ . Then for all $t\geq 0$ ,

$$ \begin{align*} \mathbb{P}_{\mathfrak{x}}\left(|a^*\mathfrak{x}|\geq t\right)\leq 2\mathrm{e}^{-\frac{t^2}{11 K^2}}. \end{align*} $$

In particular, $a^*\mathfrak {x}$ is a subgaussian random variable with $\psi _2$ -norm at most $5K$ .

Proposition 4.26 (Anti-concentration bound).

Let $\mathfrak {x}\in \mathbb {R}^N$ be a random vector such that its components $\mathfrak {x}_i$ are independent random variables with anti-concentration property with constant $\rho $ . Then for every $A\in \mathbb {R}^{k\times N}$ with rank k and measurable $U\subseteq \mathbb {R}^k$ ,

$$ \begin{align*}\mathbb{P}_{\mathfrak{x}}\left(A\mathfrak{x}\in U\right)\leq \frac{\operatorname{\mathrm{vol}}(U)(\sqrt{2}\rho)^k}{\sqrt{\det\left(AA^*\right)}}.\end{align*} $$

Proof of Proposition 4.24.

This is just [Reference Vershynin58, Proposition 2.5.2] with improved constants.

For the first part, we give a proof since we don’t explicitly use the constants in the proof of [Reference Vershynin58, Proposition 2.5.2]. Fix $\lambda>0$ . Then by Markov’s inequality and expanding the exponential as a power series,

$$ \begin{align*} \mathbb{P}(|\mathfrak{x}|\geq t) = \mathbb{P}\left(\mathrm{e}^{\lambda^2\mathfrak{x}^2}\geq \mathrm{e}^{\lambda^2t^2}\right) \leq \mathrm{e}^{-\lambda^2t^2}\sum_{p=0}^\infty \frac{\lambda^{2p}\mathop{\mathbb{E}} \mathfrak{x}^{\,2p}}{p!} \leq \mathrm{e}^{-\lambda^2t^2}\sum_{p=0}^\infty \frac{(\lambda^22pK^2)^p}{p!}. \end{align*} $$

Now, by setting the value of $\lambda $ to $\frac {1}{\sqrt {6}K}$ , $ \mathbb {P}(|\mathfrak {x}|\geq t) \leq \mathrm {e}^{-\frac {t^2}{8K^2}}\sum _{p=0}^\infty \frac {(p/3)^p}{p!}. $ The right-hand series is convergent, and after adding the series numerically, we can see that $ \sum _{p=0}^\infty \frac {(p/3)^p}{p!}= 2.625\ldots \leq \mathrm {e}, $ which finishes the proof of the first part. Following the constants in the proof of [Reference Vershynin58, Proposition 2.5.2] directly seems to give $4\mathrm {e}\simeq 10.8$ in the denominator of the exponent instead of $6$ .

For the second, note that

$$ \begin{align*} \mathop{\mathbb{E}}|\mathfrak{x}|^p=K^p\left(2\ln C\right)^{\frac{p}{2}}+\int_0^\infty pu^{p-1}\mathrm{e}^{-\frac{u^2}{2K}}\,\mathrm{d}u, \end{align*} $$

which follows from

$$ \begin{align*} \mathbb{P}(|\mathfrak{x}|>u)\leq \begin{cases}1&\text{if }u\leq K\sqrt{2\ln C}\\ \mathrm{e}^{-\frac{u^2}{2K^2}}&\text{if }u\geq K\sqrt{2\ln C}, \end{cases} \end{align*} $$

dividing the integration domain into $[0,K\sqrt {2\ln C}]$ and $[K\sqrt {2\ln C},\infty ]$ and applying some straightforward calculations and bounds.

Now, applying the change of variables $t=\frac {u^2}{2K}$ , we obtain

$$ \begin{align*} \int_0^\infty pu^{p-1}\mathrm{e}^{-\frac{u^2}{2K}}\,\mathrm{d}u =pK^p2^{\frac{p}{2}-1}\Gamma\left(\frac{p}{2}\right) \leq K^p\left(\frac{\pi p}{2}\right)^{\frac{p}{2}}. \end{align*} $$

Hence

$$ \begin{align*} \mathop{\mathbb{E}}|\mathfrak{x}|^p\leq K^p\left(\left(2\ln C\right)^{\frac{p}{2}}+\left(\frac{\pi p}{2}\right)^{\frac{p}{2}}\right), \end{align*} $$

from which the second part follows.

Proof of Proposition 4.25.

This is a version of [Reference Vershynin58, Proposition 2.6.1]. Let us sketch a proof to see the values of the chosen constants.

Let $\mathfrak {y}\in \mathbb {R}$ be a centred random variable with $\psi _2$ -norm at most K. Arguing as in part ‘ii $\Rightarrow $ iii’ of the proof of [Reference Vershynin58, Proposition 2.5.2], we have that for all $\lambda \in [-1/\sqrt {2\mathrm {e}},1/\sqrt {2\mathrm {e}}]$ ,

$$ \begin{align*}\mathop{\mathbb{E}} \mathrm{e}^{\lambda^2\mathfrak{y}^2}\leq \mathrm{e}^{\mathrm{e} K^2\lambda^2},\end{align*} $$

using $n!\geq \sqrt {2\pi }(n/e)^n$ and that for $x\in [-1/2,1/2]$ , we have $1+\frac {1}{\sqrt {2\pi }}\frac {x^2}{1-x^2}\leq \mathrm {e}^{x^2/2}$ . Then arguing as in part ‘iii $\Rightarrow $ v’ of the proof of [Reference Vershynin58, Proposition 2.5.2], we get that for all $\lambda \in \mathbb {R}$ ,

(4.5) $$ \begin{align} \mathop{\mathbb{E}}\mathrm{e}^{\lambda\mathfrak{y}}\leq \mathrm{e}^{\mathrm{e} K^2\lambda^2}. \end{align} $$

In this way, we have that

$$ \begin{align*} \mathbb{P}(|a^*\mathfrak{x}|\geq t)&\leq 2\mathbb{P}(a^*\mathfrak{x}\geq t)&\text{(symmetry)}\\ &=2\mathbb{P}(\mathrm{e}^{a^*\mathfrak{x}}\geq \mathrm{e}{t})&\\ &\leq 2\mathrm{e}^{-\lambda t}\mathop{\mathbb{E}}\mathrm{e}^{\lambda a^*\mathfrak{x}}&\text{(Markov's inequality)}\\ &= 2\mathrm{e}^{-\lambda t}\prod_{i=1}^N\mathop{\mathbb{E}}\mathrm{e}^{\lambda a_i\mathfrak{x}_i}&\text{(}a_1\mathfrak{x}_1,\ldots,a_N\mathfrak{x}_N\text{ independent)}\\ &\leq 2\mathrm{e}^{-\lambda t}\prod_{i=1}^N\mathrm{e}^{\mathrm{e} a_i^2K^2\lambda^2}&(4.5)\\ &= 2\mathrm{e}^{-\lambda t+\mathrm{e} K^2\lambda^2}&(\|a\|_2=1). \end{align*} $$

Taking $\lambda =\frac {t}{2\mathrm {e} K^2}$ , we get the desired tail bound. The last claim immediately follows from Proposition 4.24.

Proof of Proposition 4.26.

This is a rewriting of [Reference Rudelson and Vershynin51, Theorem 1.1] using [Reference Livshyts, Paouris and Pivovarov44] to get explicit constants. This rewriting was first given in [Reference Tonelli-Cueto and Tsigaridas56, Proposition 2.5]. We provide the argument for the sake of completeness.

By the SVD, we have $A=P\Sigma Q$ , where P is an isometry, $\Sigma \in \mathbb {R}^{k\times k}$ a positive diagonal matrix and Q an orthogonal projection. Hence

$$ \begin{align*} \mathbb{P}_{\mathfrak{x}}\left(A\mathfrak{x}\in U\right)=\mathbb{P}_{\mathfrak{x}}\left(Q\mathfrak{x}\in \Sigma^{-1}P^*U\right), \end{align*} $$

and since $\operatorname {\mathrm {vol}}(\Sigma ^{-1}P^*U)=\operatorname {\mathrm {vol}}(U)/\det \Sigma =\operatorname {\mathrm {vol}}(U)/\sqrt {\det (AA^*)}$ , we only have to prove the claim for the case in which A is an orthogonal projection.

Now, by [Reference Rudelson and Vershynin51, Theorem 1.1] (see [Reference Livshyts, Paouris and Pivovarov44, Theorem 1.1] for getting the constant), we have that $A\mathfrak {x}$ has density bounded by $\sqrt {2}\rho $ . Thus $\mathbb {P}(A\mathfrak {x}\in U)\leq \operatorname {\mathrm {vol}}(U)(\sqrt {2}\rho )^k$ , as we wanted to show.

4.3.2 $\mathsf {K}$ vs. $\kappa $ : Measuring the effect of the $L_\infty $ -norm on the grid method

The condition-based complexity estimates we obtained in this section essentially substitute the $\kappa $ in the cost estimates of the original algorithm by $\mathsf {K}$ . This way, the comparison between the two algorithms reduces to estimate $\mathsf {K}/\kappa $ . The following proposition shows that, in turn, this amounts to looking at the quotient $\|f\|_\infty ^{\mathbb {R}}/\|f\|_W$ .

Proposition 4.27. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $x\in \mathbb {S}^n$ . Then

$$ \begin{align*} \frac{\|f\|_\infty^{\mathbb{R}}}{\|f\|_W}\leq \frac{\mathsf{K}(f,x)}{\kappa(f,x)}\leq \sqrt{2q\mathbf{D}}\frac{\|f\|_\infty^{\mathbb{R}}}{\|f\|_W} \end{align*} $$

and

$$ \begin{align*} \frac{\|f\|_\infty^{\mathbb{R}}}{\|f\|_W}\leq \frac{\mathsf{K}(f)}{\kappa(f)}\leq \sqrt{2q\mathbf{D}}\frac{\|f\|_\infty^{\mathbb{R}}}{\|f\|_W} .\end{align*} $$

Proof. It follows from

$$ \begin{align*} \frac{\mathsf{K}(f,x)}{\kappa(f,x)}=\sqrt{q}\frac{\|f\|_\infty^{\mathbb{R}}}{\|f\|_W}\frac{\sqrt{\|f(x)\|^2+\sigma_q\left(\Delta^{-\frac{1}{2}}\mathrm{D}_xf\right)^2}}{\max\left\{\|f(x)\|,\sigma_q\left(\Delta^{-1}\mathrm{D}_xf\right)\right\}} \end{align*} $$

and

$$ \begin{align*} \frac{1}{\sqrt{\mathbf{D}}}\;\sigma_q\left(\Delta^{-\frac{1}{2}}\mathrm{D}_xf\right) \leq \sigma_q\left(\Delta^{-1}\mathrm{D}_xf\right) \leq \sigma_q\left(\Delta^{-\frac{1}{2}}\mathrm{D}_xf\right).\\[-44pt] \end{align*} $$

In general, we have that $\frac {\|f\|_\infty ^{\mathbb {R}}}{\|f\|_W}\leq 1$ , so the corresponding quotient of condition numbers worsens by a factor of at most $\sqrt {2q\mathbf {D}}$ . Our main result derives from the fact that $\frac {\|f\|_\infty ^{\mathbb {R}}}{\|f\|_W}$ is, for a substantial number of fs, much smaller than 1: we can expect it to be smaller than $\sqrt {n\ln (\mathrm {e}\mathbf {D})/N}$ with very high probability. Recall that $K \rho $ is a constant from the randomness model.

Theorem 4.28. Let $q\leq n+1$ , $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ be dobro with parameters K and $\rho $ and $\ell \in \mathbb {N}$ . For any power $\ell $ with $1 \leq \ell < \frac {N}{2}$ , we have

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{f}}\left(\frac{\|\mathfrak{f}\|_\infty^{\mathbb{R}}}{\|\mathfrak{f}\|_W}\right)^\ell \leq \left(\frac{890\sqrt{2}K\rho\sqrt{n\ln(eD)\ell}}{\sqrt{N-2\ell}}\right)^\ell. \end{align*} $$

In particular,

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{f}} \frac{\|\mathfrak{f}\|_\infty^{\mathbb{R}}}{\|\mathfrak{f}\|_W} \leq \mathcal{O}\left(K\rho\sqrt{ \frac{n \ln(e \mathbf{D})}{N}} \right). \end{align*} $$

Remark 4.29. In the study of tensors, the quotients $\|\mathfrak {f}\|_\infty ^{\mathbb {R}}/\|\mathfrak {f}\|_W$ and their nonsymmetric analogue play an important role. Because of this, we can consider Theorem 4.28 a symmetric analogue of the results shown in [Reference Gross, Flammia and Eisert38] and [Reference Nguyen, Drineas and Tran46]. In a paper under preparation by Kozhasov and the third author [Reference Kozhasov and Tonelli-Cueto41], the probabilistic techniques introduced in this paper are developed further to study $\|\mathfrak {f}\|_\infty ^{\mathbb {R}}/\|\mathfrak {f}\|_W$ in several settings.

Corollary 4.30. Let $q\leq n+1$ and $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ be dobro with parameters K and $\rho $ . Then for $1 \leq \ell < \frac {N}{2}$ , we have

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{f}}\left(\frac{\mathsf{K}(\mathfrak{f})}{\kappa(\mathfrak{f})}\right)^\ell \leq \left(\frac{1780}K\rho\sqrt{qn\mathbf{D}\ln(eD)\ell}{\sqrt{N-2\ell}}\right)^\ell. \end{align*} $$

Let PolyBetti ${}_W$ be the version of

using the Weyl norm and $\kappa $ . An analysis along the lines of [Reference Cucker, Krick and Shub28] (or [Reference Bürgisser, Cucker and Lairez15]) shows that the cost of PolyBetti ${}_W$ is

$$ \begin{align*}2^{\mathcal{O}(n^2\log n)}\mathbf{D}^{10 n}\kappa(f)^{10 n},\end{align*} $$

which is very similar to the cost bound for

in Proposition 4.14. Let us denote by

and

these cost bounds. It follows that

Using Corollary 4.30 and Markov’s inequality, it is easy to prove the following estimate.

Corollary 4.31. Let $q\leq n+1$ , $N>20n$ and $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ be dobro with parameters K and $\rho $ ,

with probability at least $1-1/N$ . Note that for fixed n and large $\mathbf {D}$ , the ratio in the right-hand side is of the order of

$$ \begin{align*} \left( \frac{ K\rho\sqrt{\ln(\mathrm{e}\mathbf{D})}}{\mathbf{D}}^{\frac{n-1}{2}} \right)^{10n}. \end{align*} $$

We proceed to prove Theorem 4.28.

Proposition 4.32. Let $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ be dobro with parameters K and $\rho $ . Then for all $t>0$ ,

$$ \begin{align*} \mathbb{P}\left(\|\mathfrak{f}\|_\infty^{\mathbb{R}}\geq t\right) \leq q\sqrt{2\pi}\sqrt{n+1}\left(\frac{\mathrm{e} \mathbf{D}}{2}\right)^n\mathrm{e}^{-\frac{t^2}{17K^2}}. \end{align*} $$

In particular, if $q\leq n+1$ , for all $\ell \geq 1$ , $\left (\mathop {\mathbb {E}}_{\mathfrak {f}}\left (\|\mathfrak {f}\|_\infty ^{\mathbb {R}}\right )^\ell \right )^{\frac {1}{\ell }} \leq 63K\sqrt {n\ln (\mathrm {e}\mathbf {D})\ell }$ .

Proof of Theorem 4.28.

By the Cauchy-Schwarz inequality,

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{f}}\left(\frac{\|\mathfrak{f}\|_\infty^{\mathbb{R}}}{\|\mathfrak{f}\|_W}\right)^\ell &\leq \sqrt{\mathop{\mathbb{E}}_{\mathfrak{f}}\left(\|\mathfrak{f}\|_\infty^{\mathbb{R}}\right)^{2\ell}} \sqrt{\mathop{\mathbb{E}}_{\mathfrak{f}}\frac{1}{\|\mathfrak{f}\|_W^{2\ell}}}. \end{align*} $$

The first term on the right is bounded by Proposition 4.32.

For the second term, we will use [Reference Mendelson and Paouris45, Theorem 1.11]. We note that $\mathfrak {x}\in \mathbb {R}^N$ satisfies the small ball assumption (SBA) with constant $\mathcal {L}$ [Reference Mendelson and Paouris45, Assumption 1.1.] if for every $k\in \{1,\ldots ,N-1\}$ , every orthogonal projection $P\in \mathbb {R}^{k\times N}$ , every $y\in \mathbb {R}^k$ and every $\varepsilon>0$ ,

$$ \begin{align*} \mathbb{P}\left(\|P\mathfrak{x}-y\|_2\leq \sqrt{k}\varepsilon\right) \leq (\mathcal{L}\varepsilon)^k. \end{align*} $$

By Proposition 4.26 (applied with coordinates orthogonal with respect to the Weyl inner product) and Stirling’s approximation, we have that $\mathfrak {f}$ has the SBA with constant $2\sqrt {\pi e}\rho $ . Thus, by [Reference Mendelson and Paouris45, Theorem 1.11],

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{f}}\frac{1}{\|\mathfrak{f}\|_W^{2\ell}}\leq (14\rho)^{2\ell} \mathop{\mathbb{E}}_{\mathfrak{g}}\frac{1}{\|\mathfrak{g}\|_W^{2\ell}}, \end{align*} $$

where $\mathfrak {g}\in \mathcal {H}_{\boldsymbol {d}}[q]$ is KSS. Since $\mathfrak {g}$ is a Gaussian vector for all coordinate systems orthogonal with respect to the Weyl inner product, $\|\mathfrak {g}\|_W^{2}$ is distributed according to a $\chi ^2$ -distribution with N degrees of freedom. Therefore,

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{g}}\frac{1}{\|\mathfrak{g}\|_W^{2\ell}} =\int_{0}^\infty t^{-\ell}\frac{1}{2^{\frac{N}{2}} \Gamma\left(\frac{N}{2}\right)}t^{\frac{N}{2}-1} \mathrm{e}^{-\frac{t}{2}}\,\mathrm{d}t =\frac{\Gamma\left(\frac{N}{2}-\ell\right)}{2^{\ell} \Gamma\left(\frac{N}{2}\right)}=\frac{1}{(N-2)(N-4)\cdots(N-2\ell)}. \end{align*} $$

The desired claim now follows.

Proof of Proposition 4.32.

Fix $\delta \in [0,1/\mathbf {D}]$ . By the proof of Proposition 4.3, we have that $\|\mathfrak {f}\|_\infty ^{\mathbb {R}}>t$ implies $\operatorname {\mathrm {vol}}\left \{x\in \mathbb {S}^n\mid \|\mathfrak {f}(x)\|_\infty \geq \left (1-\frac {\mathbf {D}^2}{2}\delta ^2\right )t\right \} \geq \operatorname {\mathrm {vol}} B_{\mathbb {S}}(x_*,\delta )$ , where $x_*\in \mathbb {S}^n$ maximises $\|f(x)\|_\infty $ . Therefore,

$$ \begin{align*} \mathbb{P}\left(\|\mathfrak{f}\|_\infty^{\mathbb{R}}\geq t\right)& \leq \mathbb{P}_{\mathfrak{f}}\left(\mathbb{P}_{\mathfrak{x}\in\mathbb{S}^n} \left(\|\mathfrak{f}(\mathfrak{x})\|_\infty\geq \left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)t\right)\geq \operatorname{\mathrm{vol}} B_{\mathbb{S}}(x_*,\delta)/\operatorname{\mathrm{vol}}\mathbb{S}^n \right). \end{align*} $$

By [Reference Bürgisser and Cucker14, Lemma 2.25], [Reference Bürgisser and Cucker14, Lemma 2.31] and $\int _0^\delta \,n\sin ^{n-1}\theta \,\mathrm {d}\theta \geq (1-\delta ^2/6)^n\delta ^n$ , we have that

$$ \begin{align*} \operatorname{\mathrm{vol}}_n B_{\mathbb{S}}(x_*,\delta)/\operatorname{\mathrm{vol}}_n\mathbb{S}^n\geq \frac{\left(1-\delta^2/6\right)^n}{\sqrt{2\pi}\sqrt{n+1}}\delta^n. \end{align*} $$

In this way,

$$ \begin{align*} \mathbb{P}&\left(\|\mathfrak{f}\|_\infty^{\mathbb{R}}\geq t\right)\\ &\leq \mathbb{P}_{\mathfrak{f}}\left(\mathbb{P}_{\mathfrak{x}\in\mathbb{S}^n}\left(\|\mathfrak{f}(\mathfrak{x})\|_\infty \geq \left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)t\right) \geq \frac{\left(1-\delta^2/6\right)^n}{\sqrt{2\pi}\sqrt{n+1}}\delta^n\right) \\ &\leq \frac{\sqrt{2\pi}\sqrt{n+1}}{\left(1-\delta^2/6\right)^n\delta^n}\, \mathop{\mathbb{E}}_{\mathfrak{f}} \mathbb{P}_{\mathfrak{x}\in\mathbb{S}^n}\left(\|\mathfrak{f}(\mathfrak{x})\|_\infty \geq \left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)t\right)&\text{(Markov's inequality)}\\ &\leq \frac{\sqrt{2\pi}\sqrt{n+1}}{\left(1-\delta^2/6\right)^n\delta^n}\, \mathop{\mathbb{E}}_{\mathfrak{x}\in\mathbb{S}^n} \mathbb{P}_{\mathfrak{f}}\left(\|\mathfrak{f}(\mathfrak{x})\|_\infty \geq \left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)t\right)&\text{(Tonelli's theorem)}\\ &\leq \frac{\sqrt{2\pi}\sqrt{n+1}}{\left(1-\delta^2/6\right)^n\delta^n}\, \max_{x\in\mathbb{S}^n} \mathbb{P}_{\mathfrak{f}}\left(\|\mathfrak{f}(x)\|_\infty \geq \left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)t\right)\\ &\leq \frac{q\sqrt{2\pi}\sqrt{n+1}}{\left(1-\delta^2/6\right)^n\delta^n}\, \max_{i,x\in\mathbb{S}^n} \mathbb{P}_{\mathfrak{f}}\left(|\mathfrak{f}_i(x)|_\infty \geq \left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)t\right)&\text{(Union bound)}. \end{align*} $$

In the coordinates of a monomial basis orthogonal for the Weyl inner product, the following holds: (1) a dobro random polynomial $\mathfrak {f}$ looks like a random vector whose components are independent and subgaussian of $\psi _2$ -norm at most K, and (2) evaluation at a point of the sphere, $\mathfrak {f}(x)$ , becomes the inner product with a vector of norm 1 (by Proposition 2.2). Hence, by Proposition 4.24,

$$ \begin{align*} \mathbb{P}\left(\|\mathfrak{f}\|_\infty^{\mathbb{R}}\geq t\right) \leq \frac{q\sqrt{2\pi}\sqrt{n+1}}{\left(1-\delta^2/6\right)^n\delta^n}\, \exp\left(-\left(1-\frac{\mathbf{D}^2}{2}\delta^2\right)^2\frac{t^2}{11K^2}\right). \end{align*} $$

The claim follows taking $\delta =5/(6\mathbf {D})$ and $\left (1-\frac {1}{2}\left (\frac {5}{6}\right )^2\right )\frac {1}{11}\geq \frac {1}{17}$ . For the other inequalities on the moments, use Proposition 4.24.

4.3.3 Complexity of the Plantinga-Vegter algorithm

In [Reference Cucker, Ergür and Tonelli-Cueto24] (compare to [Reference Cucker, Ergür and Tonelli-Cueto23]), we proved the following result (which we are just adapting to the notationFootnote ⁵ of this paper).

Theorem 4.33 [Reference Cucker, Ergür and Tonelli-Cueto24, Theorem 8.4 and Theorem 7.3].

Let $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[1]$ be dobro with parameters K and $\rho $ . For all $x\in \mathbb {S}^n$ and $t\geq e$ ,

$$ \begin{align*} \mathbb{P} \left(\kappa(\mathfrak{f},x) \geq t\right) \leq 2 \left(\frac{N} {n+1}\right)^{\frac{n+1}{2}}(30 K\rho)^{n+1} \frac{\ln^{\frac{n+1}{2}}t}{t^{n+1}}. \end{align*} $$

In particular, for the Plantinga-Vegter algorithm with input $\mathfrak {f}$ over the domain $[-a,a]^n$ , the expected number of hypercubes in the final subdivision is at most

$$ \begin{align*} a^n \mathbf{D}^{n} N^{\frac{n+1}{2}} 2^{n \log n + 13n + \frac{3}{2}\log n+\frac{17}{2}} (K\rho)^{n+1}. \end{align*} $$

Our objective is the following theorem, which shows how the $N^{\frac {n+1}{2}}$ factor vanishes from these estimates when we pass from $\kappa $ to $\mathsf {K}$ . This shows that the version of Plantinga-Vegter using $\mathsf {K}$ yields better cost bounds than the one using $\kappa $ : that is, the one in [Reference Cucker, Ergür and Tonelli-Cueto24].

Theorem 4.34. Let $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[1]$ be dobro with parameters K and $\rho $ . For all $x\in \mathbb {S}^n$ and $t\geq e$ ,

$$ \begin{align*} \mathbb{P} \left(\mathsf{K}(\mathfrak{f},x) \geq t\right) \leq \mathbf{D}^{\frac{n}{2}}(\ln \mathrm{e}\mathbf{D})^{\frac{n+1}{2}}2^{6n+4}(K\rho)^{n+1} \frac{\ln^{\frac{n+1}{2}}t}{t^{n+1}}. \end{align*} $$

It follows that for every compact $\Omega \subseteq \mathbb {S}^n$ ,

$$ \begin{align*} \mathop{\mathbb{E}}_{\mathfrak{f}}\mathop{\mathbb{E}}_{\mathfrak{x}\in \Omega}\left(\mathsf{K}(\mathfrak{f},\mathfrak{x})^{n}\right) \leq \mathbf{D}^{\frac{n}{2}}(\ln \mathrm{e}\mathbf{D})^{\frac{n+1}{2}} 2^{\frac{1}{2}n\log n+5n+2\log(n)+7} (K\rho)^{n+1}. \end{align*} $$

In particular, for the Plantinga-Vegter algorithm with input $\mathfrak {f}$ over the domain $[-a,a]^n$ , the expected number of hypercubes in the final subdivision is at most

$$ \begin{align*} a^n \mathbf{D}^{\frac{3n}{2}}(\ln \mathrm{e}\mathbf{D})^{\frac{n+1}{2}} 2^{\frac{3}{2}n\log n+13n+2\log(n)+7}. \end{align*} $$

Remark 4.35. Theorem 4.34 allows us to compare the efficiency of Plantinga-Vegter for the versions based on the Weyl-norm and the $\infty $ -norm. One can observe that (in the region of interest $\mathbf {D}> n$ ) the term $N^{\frac {n}{2}} \sim \mathbf {D}^{\frac {n^2}{2}}$ in the estimate for the Weyl-norm version is replaced with $(\mathbf {D} \log \mathbf {D})^{\frac {n}{2}} $ in the $\infty $ -norm. Basically, the exponent of $\mathbf {D}$ goes from $\mathcal {O}(n^2)$ to $\mathcal {O}(n)$ . If we focus on the original cases of interest (compare to [Reference Plantinga and Vegter49]) – that is, $n=2$ and $n=3$ , with the average complexity analysis from [Reference Cucker, Ergür and Tonelli-Cueto24] – it is shown in Theorem 3.1 there that PV-Interval ${}_W$ has an average complexity of

$$ \begin{align*} \mathcal{O}\left(d^8\max\{1,a^2\}(K\rho)^3\right) &\qquad \text{for}\ n=2,\ \text{and}\\ \mathcal{O}\left(d^{13}\max\{1,a^3\}(K\rho)^4\right) &\qquad \text{for}\ n=3. \end{align*} $$

It follows from Theorems 4.19 and 4.34 that the average complexity of PV-Interval ${}_\infty $ is

$$ \begin{align*} \mathcal{O}\left(d^7\log^{1.5}(d)\max\{1,a^2\}(K\rho)^3\right) &\qquad \text{for}\ n=2,\ \text{and}\\ \mathcal{O}\left(d^{10}\log^2(d)\max\{1,a^3\}(K\rho)^4\right) &\qquad \text{for}\ n=3. \end{align*} $$

We next proceed to prove Theorem 4.34.

Proof of Theorem 4.34.

Let $u,t\geq 0$ , then

$$ \begin{align*} \mathbb{P}_{\mathfrak{f}}&\left(\mathsf{K}(\mathfrak{f},x)\geq t\right)\\ &\leq \mathbb{P}_{\mathfrak{f}}\left(\|f\|_\infty^{\mathbb{R}}\geq u\text{ or }\max\left\{|\mathfrak{f}(x)|,\frac{\|\mathrm{D}_x\mathfrak{f}\|}{\mathbf{D}}\right\}\leq \frac{u}{t}\right)&\text{(implication bound)}\\ &\leq \mathbb{P}_{\mathfrak{f}}\left(\|f\|_\infty^{\mathbb{R}}\geq u\right)+\mathbb{P}_{\mathfrak{f}}\left(\max\left\{|\mathfrak{f}(x)|,\frac{\|\mathrm{D}_x\mathfrak{f}\|}{\mathbf{D}}\right\}\leq \frac{u}{t}\right),&\text{(union bound)} \end{align*} $$

where we used the fact that for $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[1]$ , $\mathsf {K}(f,x)=\|f\|_\infty ^{\mathbb {R}}/\max \left \{|f(x)|,\|\mathrm {D}_xf\|/\mathbf {D}\right \}$ .

On the one hand, $\mathbb {P}_{\mathfrak {f}}\left (\|f\|_\infty ^{\mathbb {R}}\geq u\right )$ is bounded by Proposition 4.32. On the other hand, the map

$$ \begin{align*} f\mapsto \begin{pmatrix} f(x)&\frac{\mathrm{D}_xf}{\mathbf{D}} \end{pmatrix} \end{align*} $$

has singular values $1,1/\sqrt {\mathbf {D}},\ldots ,1/\sqrt {\mathbf {D}}$ in the coordinates of a monomial basis orthogonal with respect to the Weyl inner product. And since in such a basis, a dobro polynomial is a vector whose coefficients are independent and have the anti-concentration property with constant $\rho $ , we deduce that

$$ \begin{align*} & \mathbb{P}_{\mathfrak{f}}\left(\max\left\{|\mathfrak{f}(x)|,\frac{\|\mathrm{D}_x\mathfrak{f}\|}{\mathbf{D}}\right\} \leq \frac{u}{t}\right)\leq \mathbf{D}^{\frac{n}{2}}\operatorname{\mathrm{vol}}\left\{(x_0,x)\in\mathbb{R}^{n+1}\mid |x_0|,\|x\| \leq u/t\right\}\left(\sqrt{2}\rho\right)^{n+1}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \leq \omega_n\mathbf{D}^{\frac{n}{2}}\left(\frac{\sqrt{2}\rho u}{t}\right)^{n+1} \leq 9^n\mathbf{D}^{\frac{n}{2}}\left(\frac{u\rho}{\sqrt{n}}\right)^{n+1} \frac{1}{t^{n+1}}, \end{align*} $$

where $\omega _n$ is the volume of the unit n-ball, and we used Proposition 4.26 and Stirling’s estimation [Reference Bürgisser and Cucker14, Equation (2.14)].

Hence, combining the inequalities above,

$$ \begin{align*} \mathbb{P}_{\mathfrak{f}}\left(\mathsf{K}(\mathfrak{f},x)\geq t\right)\leq \sqrt{2\pi(n+1)}\left(\frac{\mathrm{e}\mathbf{D}}{2}\right)^n\mathrm{e}^{-\frac{u^2}{17K^2}} +9^n\mathbf{D}^{\frac{n}{2}}\left(\frac{u\rho}{\sqrt{n}}\right)^{n+1} \frac{1}{t^{n+1}}. \end{align*} $$

Taking $t\geq e$ and $u=\sqrt {17}K\sqrt {n\ln (\mathrm {e}^2\mathbf {D})\ln t}\geq \sqrt {17}K\sqrt {n\ln \mathbf {D}+(n+1)\ln t}$ , we get

$$ \begin{align*} \mathbb{P}_{\mathfrak{f}}\left(\mathsf{K}(\mathfrak{f},x)\geq t\right)\leq \frac{\sqrt{2\pi(n+1)}}{(2\mathrm{e})^n}\frac{1}{t^{n+1}}+9^n\mathbf{D}^{\frac{n}{2}}\left(\sqrt{17}K\rho\ln^{\frac{1}{2}(\mathrm{e}^2 \mathbf{D})}\right)^{n+1}\frac{\ln^{\frac{n+1}{2}}t}{t^{n+1}}. \end{align*} $$

This proves the first statement.

By Tonelli’s theorem, to prove the second statement, it is enough to bound $\mathop {\mathbb {E}}_{\mathfrak {f}}\mathsf {K}(\mathfrak {f},x)^n$ for a fixed $x\in \mathbb {S}^n$ . Now,

$$ \begin{align*} &\mathop{\mathbb{E}}_{\mathfrak{f}}\mathsf{K}(\mathfrak{f},x)^n=\int_{0}^\infty\mathbb{P}_{\mathfrak{f}}\left(\mathsf{K}(\mathfrak{f},x) \geq t^{\frac{1}{n}}\right)\\ &\qquad\qquad\le \mathrm{e}^n+\int_{\mathrm{e}^n}^\infty\mathbb{P}_{\mathfrak{f}} \left(\mathsf{K}(\mathfrak{f},x)\geq t^{\frac{1}{n}}\right) \leq \mathrm{e}^n+\int_{\mathrm{e}^n}^\infty \mathbf{D}^{\frac{n}{2}}\ln^{\frac{n+1}{2}}(\mathrm{e}\mathbf{D})2^{6n+4}(K\rho)^{n+1} \frac{\ln(t^{\frac{1}{n}})^{\frac{n+1}{2}}}{t^{1+\frac{1}{n}}}\, \mathrm{d}t\\ &\qquad\qquad\qquad\qquad\qquad\qquad\leq \mathrm{e}^n+\mathbf{D}^{\frac{n}{2}}\ln^{\frac{n+1}{2}}(\mathrm{e}\mathbf{D})2^{6n+4}(K\rho)^{n+1} \int_{1}^\infty\frac{\ln(t^{\frac{1}{n}})^{\frac{n+1}{2}}} {t^{1+\frac{1}{n}}}\,\mathrm{d}t. \end{align*} $$

By changing variables, $t=\mathrm {e}^{sn}$ , we can see that

$$ \begin{align*} \int_{1}^\infty\frac{\ln(t^{\frac{1}{n}})^{\frac{n+1}{2}}} {t^{1+\frac{1}{n}}}\,\mathrm{d}t =n\Gamma\left(\frac{n+3}{2}\right) \leq \sqrt{2\pi\mathrm{e}}\,n\sqrt{n+1} \left(\frac{n+1}{2\mathrm{e}}\right)^{\frac{n+1}{2}}, \end{align*} $$

where the inequality comes from Stirling’s approximation [Reference Bürgisser and Cucker14, Equation (2.14)]. Hence, we get

$$ \begin{align*} &\mathop{\mathbb{E}}_{\mathfrak{f}}\mathsf{K}(\mathfrak{f},x)^n\leq \mathrm{e}^n+\sqrt{2\pi\mathrm{e}}n\sqrt{n+1}\mathbf{D}^{\frac{n}{2}} \ln^{\frac{n+1}{2}}(\mathrm{e}\mathbf{D})\frac{2^{6n+4}}{(2\mathrm{e})^{\frac{n+1}{2}}}(\sqrt{n+1}K\rho)^{n+1}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq 8n\sqrt{n+1}\mathbf{D}^{\frac{n}{2}}\ln^{\frac{n+1}{2}} (\mathrm{e}\mathbf{D})\frac{2^{6n+4}}{(2\mathrm{e})^{\frac{n+1}{2}}}(\sqrt{n+1}K\rho)^{n+1}. \end{align*} $$

The second statement now follows after some easy bounds.

5.1 Description of the linear homotopy

The algorithm below is, essentially, the one in [Reference Bürgisser and Cucker13] and [Reference Bürgisser and Cucker14, Chapter 17]. The only change is in the computation of the step-length $\Delta _t$ , where we replace the original (here $\mathrm {dist}_{\mathbb {S}}$ denotes angle)

$$ \begin{align*}\frac{0.008535284}{\mathrm{dist}_{\mathbb{S}}(f,g) \mathbf{D}^{3/2}\mu_{\mathrm{norm}}^2(q,z)} \end{align*} $$

by

(5.1) $$ \begin{align} \frac{0.03\,\|q\|_{\infty}^{\mathbb{C}}} {\|f-g\|_{\infty}^{\mathbb{C}}\mathbf{D}\mathsf{M}^2(q,z)}. \end{align} $$

This change amounts – leaving aside the difference in the constants and a smaller exponent in $\mathbf {D}$ – to the use of the $\infty $ -norm instead of the Weyl one and, consequently, the use of $\mathsf {M}$ instead of $\mu _{\mathrm {norm}}$ . Recall that $N_q$ is the Newton operator associated to $q\in \mathcal {H}_{\boldsymbol {d}}[n]$ .

5.2 A bound on the number of iterations

The analysis of ALH $_\infty $ closely follows the steps in [Reference Bürgisser and Cucker14]. It uses the properties of $\mathsf {M}$ shown in Theorem 3.5 and one more result (we know for $\mu _{\mathrm {norm}}$ ): namely, that $\mathsf {M}$ is a condition number in the standard sense of this expression – it measures how solutions change when data is perturbed (see Proposition 5.4 below). To simplify the notation, in the rest of this section, we will often omit the reference to the base field $\mathbb {C}$ .

Theorem 5.2. Suppose that the lifting of the segment $[g,f]$ in $\mathcal {V}$ corresponding to $\zeta $ does not cut $\Sigma '$ . Then the algorithm ALH $_\infty $ stops after at most K steps with

$$ \begin{align*}K\le 1+45\, \mathbf{D}\,\|f-g\|_{\infty} \int_0^1 \frac{\mathsf{M}^2(q_t,\zeta_t)}{\|q_t\|_{\infty}} {\mathrm d}t. \end{align*} $$

The returned point z is an approximate zero of f with associated zero $\zeta _1$ .

Corollary 5.3. The bound K in Theorem 5.2 satisfies

$$ \begin{align*}K\le 1+45\, n\,\mathbf{D} \int_0^1 (\| f \|_{\infty} + \| g \|_{\infty})^2 \|\mathrm{D}_{\zeta_t}q_t^{-1}\Delta\|^2 {\mathrm d}t. \end{align*} $$

Proposition 5.4. Let $t\mapsto (f_t,\zeta _t)\in V$ be a smooth path. Then for all t,

$$ \begin{align*}\|\dot{\zeta}_t\|\leq \mathsf{M}(f_t,\zeta_t) \frac{\|\dot{f_t}\|_{\infty}}{\|f_t\|_{\infty}}. \end{align*} $$

Proof in Theorem 5.2.

The proof follows the lines of [Reference Bürgisser and Cucker14, Theorem 17.3]. We will therefore only offer a brief sketch. Set $\varepsilon :=\frac {1}{4}$ and $C=\frac {\varepsilon }{4}=\frac 1{16}$ . Let $q_t:=tf+(1-t)g$ . Also let $0<t_1<\ldots <t_K=1$ and $\zeta _0=z_0,\ldots ,z_K$ be the sequence of t-values and points in $\mathbb {P}^n$ , respectively, generated by the algorithm in its first K iterations. To simplify notation, we write $q_i$ and $\zeta _i$ instead of $q_{t_i}$ and $\zeta _{t_i}$ .

As in [Reference Bürgisser and Cucker14, Theorem 17.3], but using Proposition 3.6 in the place of [Reference Bürgisser and Cucker14, Proposition 16.2] and Theorem 3.5 in the place of [Reference Bürgisser and Cucker14, Theorem 16.1], one proves by induction the following statements for $i=0,\ldots ,K-1$ :

(a,i) $\mathrm {dist}_{\mathbb {P}}(z_i,\zeta _i)\le \frac {C}{\mathbf {D}\mathsf {M}(q_i,\zeta _i)}$

(b,i) $\frac {\mathsf {M}(q_i,z_i)}{1+\varepsilon } \le \mathsf {M}(q_i,\zeta _i) \le (1+\varepsilon )\mathsf {M}(q_i,z_i)$

(c,i) $\|q_i-q_{i+1}\|_{\infty } \le \frac {C\|q_i\|_{\infty }}{\mathbf {D}\mathsf {M}(q_i,\zeta _i)}$

(d,i) $\mathrm {dist}_{\mathbb {P}}(\zeta _i,\zeta _{i+1})\le \frac {C}{\mathbf {D}\mathsf {M}(q_i,\zeta _i)} \frac {1-\varepsilon }{1+\varepsilon }$

(e,i) $\mathrm {dist}_{\mathbb {P}}(z_i,\zeta _{i+1})\le \frac {2C}{(1+\varepsilon )\mathbf {D}\mathsf {M}(q_i,\zeta _i)}$

(f,i) $z_i$ is an approximate zero of $q_{i+1}$ with associated zero $\zeta _{i+1}$ .

By Proposition 3.6, $(\mathrm {c},i)$ , $(\mathrm {d},i)$ and our choice of C and $\varepsilon $ , we have that for all $t\in [t_i,t_{i+1}]$ ,

(5.2) $$ \begin{align} \frac{4}{5}\mathsf{M}(q_i,\zeta_i)\leq \mathsf{M}(q_t,\zeta_t)\leq \frac{5}{4}\mathsf{M}(q_i,\zeta_i). \end{align} $$

And, by the triangle inequality and $(\mathrm {b},i)$ , for $t\in [t_i,t_{i+1}]$ ,

(5.3) $$ \begin{align} \frac{\|q_t\|_{\infty}}{\|q_i\|_{\infty}} \le 1+C=\frac{17}{16}. \end{align} $$

The statement now easily follows. Consider any $i\in \{0,1,\ldots ,K-2\}$ . Then

$$ \begin{align*} \int_{t_i}^{t_{i+1}}\frac{\mathsf{M}^2(q_t,\zeta_t)}{\|q_t\|_\infty} \mathrm{d} t &\geq \frac{64}{85}\int_{t_i}^{t_{i+1}}\frac{\mathsf{M}^2(q_i,z_i)} {\|q_i\|_\infty} \mathrm{d} t=\frac{64}{85}\frac{\mathsf{M}^2(q_i,z_i)} {\|q_i\|_\infty}|t_{i+1}-t_i| & (({5.2})\ \text{and}\ ({5.3}))\\ & =\frac{64}{85}\frac{0.03}{\|f-g\|_\infty \mathbf{D}}.&\text{(choice of}\ \Delta t\text{)} \end{align*} $$

Hence

$$ \begin{align*}\int_{0}^1\frac{\mathsf{M}^2(q_t,z_t)}{\|q_t\|_{\infty}}\mathrm{d} t \ge \frac{192}{8500}\frac{K-1}{\|f-g\|_\infty \mathbf{D}} \ge \frac{K-1}{45\|f-g\|_{\infty}\mathbf{D}}, \end{align*} $$

and the result follows.

Proof of Proposition 5.4.

Recall from [Reference Bürgisser and Cucker14, Section 14.3] that the zero $\zeta _t$ is given by $\zeta _t=G(f_t)$ , where $G:U\subset \mathcal {H}_{\boldsymbol {d}}[n]\to \mathbb {P}^n$ is a local inverse of the projection $\pi _1:\mathcal {V}\to \mathcal {H}_{\boldsymbol {d}}[n]$ . Hence, for all $\dot {f_t}\in \mathcal {H}_{\boldsymbol {d}}[n]$ we have

(5.4) $$ \begin{align} \dot{\zeta_t}=\mathrm{D}_{f_t}G(\dot{f_t})=-(\mathrm{D}_{\zeta_t} f_t)^{-1}(\dot{f_t}(\zeta_t)), \end{align} $$

where the second equality is shown in the course of the proof of [Reference Bürgisser and Cucker14, Proposition 16.10]. Using this equality along with the fact that $(\mathrm {D}_{\zeta _t} f_t)^{-1}=(\mathrm {D}_{\zeta _t} f_t)^{\dagger }$ (as $q=n$ ), we deduce that

$$ \begin{align*} \|\dot\zeta_t\| &= \max_{\|\dot{f_t}\|_{\infty}=1} \|(\mathrm{D}_{\zeta_t} f_t)^{-1}(\dot{f_t}(\zeta_t))\|&(\text{By}\ ({5.4}))\\ &\le \left(\max_{\|\dot{f_t}\|_{\infty}=1}\|\dot{f_t}(\zeta_t)\|\right)\|(\mathrm{D}_{\zeta_t} f_t)^{-1}\|&\text{(operator norm inequality)}\\ &\le \sqrt{n}\left(\max_{\|\dot{f_t}\|_{\infty}=1}\|\dot{f_t}(\zeta_t)\|_\infty\right)\|(\mathrm{D}_{\zeta_t} f_t)^{-1}\|&\|~\|\leq \sqrt{n}\|~\|_\infty\\ &\le \sqrt{n}\|(\mathrm{D}_{\zeta_t} f_t)^{-1}\|&\text{(definition of }\|~\|_\infty\text{)}\\ &= \frac{\sqrt{n}\|f_t\|_{\infty}\|(\Delta^{-1}\mathrm{D}_{\zeta_t} f_t)^{-1}\|}{\|f_t\|_{\infty}} \;=\; \frac{\mathsf{M}(f_t,\zeta_t)}{\|f_t\|_{\infty}}.&\text{(definition of }\mathsf{M})\text{} \end{align*} $$

We recall that the norms where we have omitted subscripts are the usual norm in the case of vectors and the usual operator norm in the case of linear maps.

5.3 Average complexity analysis of Solve ${}_\infty $

The execution of Solve ${}_\infty $ on an input $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]$ amounts to calling ALH $_\infty $ on input $(f,\mathfrak {g},\mathfrak {z})$ , where $(\mathfrak {g},\mathfrak {z})\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]\times \mathbb {P}^n$ is a standard random pair. Consequently, the number of iterations of Solve ${}_\infty $ amounts to the number of iterations done by ALH $_\infty $ . The latter is a random variable as $(\mathfrak {g},\mathfrak {z})$ is random. We will further consider f random and bound the average complexity of Solve $_\infty $ by taking the expectation over both $(\mathfrak {g},\mathfrak {z})$ and f. Recall that a KSS complex random polynomial system $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]$ is a tuple of random polynomials

$$ \begin{align*} \left(\sum_{|\alpha|=d_1}\binom{d_1}{\alpha}^{\frac{1}{2}}\mathfrak{c}_{1,\alpha}X^\alpha,\ldots,\sum_{|\alpha|=d_n}\binom{d_n}{\alpha}^{\frac{1}{2}}\mathfrak{c}_{n,\alpha}X^\alpha\right) \end{align*} $$

such that the $\mathfrak {c}_{i,\alpha }$ are independent and identically distributed complex normal random variables of mean 0 and variance 1.

Our main result is the following.

Theorem 5.5. Let $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]$ . On input $\mathfrak {f}$ , Algorithm Solve ${}_\infty $ halts with probability 1 and performs

$$ \begin{align*} \mathcal{O}( n^3\mathbf{D}^2\ln(\mathrm{e}\mathbf{D})) \end{align*} $$

iteration steps on average.

We will use the following two results. The first is the complex version of Proposition 4.32 and has an almost identical proof. The main difference lies in the needed volume computations, as the geometry of the complex projective space $\mathbb {P}^n$ is somewhat different from that of the real sphere $\mathbb {S}^n$ . The second is a known result on random complex Gaussian matrices.

Proposition 5.7. Let $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]$ be a KSS complex random polynomial tuple. Then for all $t>0$ ,

$$ \begin{align*} \mathbb{P}\left(\|\mathfrak{f}\|_\infty^{\mathbb{C}}\geq t\right)\leq 2n\left(\frac{3\mathbf{D}}{2}\right)^{2n}\mathrm{e}^{-(t/3)^2}. \end{align*} $$

In particular, for all $\ell \geq 1$ , $ \left (\mathop {\mathbb {E}}_{\mathfrak {f}}\left (\|\mathfrak {f}\|_\infty ^{\mathbb {C}}\right )^\ell \right )^{\frac {1}{\ell }}\leq 12\sqrt {\ell \,n\,\ln (eD)} $ .

Proposition 5.8 [Reference Bürgisser and Cucker14, Proposition 4.27].

Let $\mathfrak {A}\in \mathbb {C}^{n\times (n+1)}$ be a random complex matrix whose entries are independent and identically distributed complex normal Gaussian variables. Then for all $t\geq 0$ ,

$$ \begin{align*} \mathrm{Prob} \left\{ \left\lVert {\mathfrak{A}^{\dagger}} \right\rVert \geq t \right\} \leq \frac{1}{16}\frac{n^2}{t^4}. \end{align*} $$

In particular, for $\ell \in [1,4)$ , $\left (\mathop {\mathbb {E}}_{\mathfrak {A}}\|\mathfrak {A}^{\dagger }\|^{\ell }\right )^{\frac {1}{\ell }}\leq \frac {\sqrt {n}}{2}\left (\frac {4}{4-\ell }\right )^{\frac {1}{\ell }}$ .

Proof of Theorem 5.5.

We are calling Algorithm ALH $_\infty $ with input $(\mathfrak {f},\mathfrak {g},\mathfrak {z})$ , where $\mathfrak {f}\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]$ is a KSS complex polynomial system and $(\mathfrak {g},\mathfrak {z})\in \mathcal {H}_{\boldsymbol {d}}[n]$ is a standard pair.

Let $\Sigma :=\{h\in \mathcal {H}_{\boldsymbol {d}}[n]\mid \exists \zeta \in \mathbb {P}^n \text { such that }(h,\zeta )\in \tilde {\Sigma }\}$ . By classic results in algebraic geometry, this set is a complex algebraic hypersurface, so it has real codimension $2$ . Hence, with probability one, the segment $[\mathfrak {g},\mathfrak {f}]$ does not intersect it, and for each zero $\zeta ^{(i)}$ of $\mathfrak {g}$ , we obtain a unique lifted path

$$ \begin{align*}t\mapsto (\mathfrak{q}_t,\zeta^{(i)}_t)\in\mathcal{V}.\end{align*} $$

Here, for each t, the $\zeta _t^{(i)}$ cover all the $d_1\cdots d_n$ different zeros of $\mathfrak {q}_t:=t\mathfrak {f}+(1-t)\mathfrak {g}$ . Recall that behind this lifting lies the fact that the map $\mathcal {V}\setminus \tilde {\Sigma }\mapsto \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[n]\setminus \Sigma $ , $(f,\eta )\mapsto f$ , is a regular covering map of degree $\mathcal {D}=d_1\cdots d_n$ .

In this way, the random zero $\mathfrak {z}$ of $\mathfrak {g}$ defines, following its lifted path, a zero $\mathfrak {z}_t$ of $\mathfrak {q}_t$ . Moreover, since the original $\mathfrak {z}$ is chosen uniformly from the $\mathcal {D}$ zeros of $\mathfrak {g}$ , the $\mathfrak {z}_t$ is a uniformly chosen zero of $\mathfrak {q}_t$ . Hence

$$ \begin{align*} \left(\frac{\mathfrak{q}_t}{\sqrt{t^2+(1-t)^2}},\mathfrak{z}_t\right)\in\mathcal{V} \end{align*} $$

is a standard random pair, since $\frac {\mathfrak {q}_t}{\sqrt {t^2+(1-t)^2}}$ is a KSS complex random polynomial and $\mathfrak {z}_t$ is a uniformly drawn zero of this system.

By Corollary 5.3, the expected number of iterations of Solve ${}_\infty $ with input $\mathfrak {f}$ is bounded by

(5.5) $$ \begin{align} 45n\,\mathbf{D}\int_{0}^1\,\mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})} \left((\|\mathfrak{f}\|_\infty^2+\|\mathfrak{g}\|_\infty^2)^2 \|\mathrm{D}_{\mathfrak{z}_t}\mathfrak{q}_t^{-1}\Delta\|^2\right)\,\mathrm{d}t, \end{align} $$

where we have moved the expectation inside the integral using Tonelli’s theorem. Now, by Hölder’s inequality,

(5.6) $$ \begin{align} \mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})}\left((\|\mathfrak{f}\|_\infty^2+\|\mathfrak{g}\|_\infty^2)^2 \|\mathrm{D}_{\mathfrak{z}_t}\mathfrak{q}_t^{-1}\Delta\|^2\right)\leq \left(\mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})}(\|\mathfrak{f}\|_\infty^2 +\|\mathfrak{g}\|_\infty^2)^6\right)^{\frac{1}{3}}\left(\mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})} \|\mathrm{D}_{\mathfrak{z}_t}\mathfrak{q}_t^{-1}\Delta\|^3\right)^{\frac{2}{3}}. \end{align} $$

By Proposition 5.7, we have that

$$ \begin{align*} \left(\mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})}(\|\mathfrak{f}\|_\infty^2+\|\mathfrak{g}\|_\infty^2)^6\right)^{\frac{1}{3}}=\mathcal{O}(n\ln(\mathrm{e}\mathbf{D})). \end{align*} $$

To apply the proposition, we expanded the binomial and used the fact that $\mathfrak {f}$ and $\mathfrak {g}$ are independent.

Because $(\mathfrak {q}_t/\sqrt {t^2+(1-t)^2},\mathfrak {z}_t)$ is a random standard pair, we have that

(5.7) $$ \begin{align} \mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})}\|\mathrm{D}_{\mathfrak{z}_t}\mathfrak{q}_t^{-1}\Delta\|^3 =\left(t^2+(1-t)^2\right)^{\frac{3}{2}} \mathop{\mathbb{E}}_{(\mathfrak{h},\mathfrak{y})\sim\rho_{\mathsf{std}}}\|\mathrm{D}_{\mathfrak{y}}\mathfrak{h}^{-1}\Delta\|^3. \end{align} $$

Now, since $(\mathfrak {h},\mathfrak {y})$ is a random standard pair, the matrix

$$ \begin{align*} \Delta^{-1/2}\overline{\mathrm{D}}_{\mathfrak{y}}\mathfrak{h}\in\mathbb{C}^{n\times (n+1)} \end{align*} $$

is a random complex Gaussian matrix. This is the so-called Beltrán-Pardo trick [Reference Bürgisser and Cucker14, Proposition 17.21(a)]. Moreover, $\|\mathrm {D}_{\mathfrak {y}}\mathfrak {h}^{-1}\Delta ^{\frac {1}{2}}\| =\|\overline {\mathrm {D}}_{\mathfrak {y}}\mathfrak {h}^{\dagger }\Delta ^{\frac {1}{2}}\|$ , since $\mathfrak {y}$ is a zero of $\mathfrak {h}$ and $\mathrm {D}_{\mathfrak {y}}\mathfrak {h}$ is just $\overline {\mathrm {D}}_{\mathfrak {y}}\mathfrak {h}$ restricted to the orthogonal complement of $\mathfrak {y}$ , which we can view as $\mathrm {T}_{\mathfrak {y}}\mathbb {P}^n$ . Because of this, by Proposition 5.8,

$$ \begin{align*} \mathop{\mathbb{E}}_{(\mathfrak{h},\mathfrak{y})\sim\rho_{\mathsf{std}}}\|\mathrm{D}_{\mathfrak{z}_t}\mathfrak{q}_t^{-1}\Delta\|^3 \leq \mathbf{D}^{\frac{3}{2}}\mathop{\mathbb{E}}_{(\mathfrak{h},\mathfrak{y})\sim\rho_{\mathsf{std}}} \left\|\left(\Delta^{-\frac{1}{2}}\overline{\mathrm{D}}_{\mathfrak{z}_t} \mathfrak{q}_t\right)^{\dagger}\right\|^3\leq \frac{1}{2}\mathbf{D}^{\frac{3}{2}}n^{\frac{3}{2}}. \end{align*} $$

Hence, integrating equation (5.7),

(5.8) $$ \begin{align} \left(\int_0^1\,\mathop{\mathbb{E}}_{(\mathfrak{f},\mathfrak{g},\mathfrak{z})}\|\mathrm{D}_{\mathfrak{z}_t}\mathfrak{q}_t^{-1}\Delta\|^3 \,\mathrm{d}t\right)^{\frac{2}{3}}=\mathcal{O}(n\,\mathbf{D}). \end{align} $$

Putting together equations (5.5), (5.6) and (5.8), the desired result follows.

5.4 Systems of quadratic equations

Theorem 5.5 is an improvement over the average number of iterations of Solve ${}_W$ , which is $\mathcal {O}(nDN)$ . Furthermore, in the case of quadratic systems, we can compute each iteration with low cost, ensuring that the average total complexity remains smaller than the one for Solve ${}_W$ , which is $\mathcal {O}(n^7)$ . The major task left, unsurprisingly, is to compute $\|q\|_\infty ^{\mathbb {C}}$ in equation (5.1). But we can use that, for a quadratic polynomial $q_i$ , we can write $q_i(X)$ as $X^TA_iX$ with $A_i$ complex symmetric and that $\|q_i\|_\infty =\|A_i\|$ . We can then compute for a quadratic system $q\in \mathcal {H}_{\mathbf {2}}[n]$ the norm $\|q\|_\infty =\max \|q_i\|_{\infty }$ . A naive approach to compute each $\|q_i\|_{\infty }$ leads to an $\mathcal {O}(n^4)$ cost for the computation of $\|q\|_{\infty }$ as it uses $\mathcal {O}(n^3)$ operations to compute each $\|q_i\|_\infty $ . Proposition 5.10 below shows we can do better. All in all, we obtain the following result.