Skip to main content Accessibility help
×
Home
Hostname: page-component-5959bf8d4d-9w8k4 Total loading time: 3.257 Render date: 2022-12-07T09:30:46.831Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue false

Functional norms, condition numbers and numerical algorithms in algebraic geometry

Published online by Cambridge University Press:  22 November 2022

Felipe Cucker
Affiliation:
Department of Mathematics, City University of Hong Kong, Hong Kong; E-mail: macucker@cityu.edu.hk
Alperen A. Ergür
Affiliation:
Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX, United States; E-mail: alperen.ergur@utsa.edu
Josué Tonelli-Cueto
Affiliation:
OURAGAN team, Inria Paris & Institut de mathématiques de Jussieu-Paris Rive Gauche, Paris, France; E-mail: josue.tonelli.cueto@bizkaia.eu

Abstract

In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand to optimise accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty }$ . This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty }$ -norm, the use of $L_p$ -norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale’s 17th problem).

Type
Computational Mathematics
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (https://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

In numerical analysis, it matters how we measure errors. Change the metric we measure the perturbations with, and a well-conditioned input may turn badly conditioned (a remarkable example is in [Reference Cheung and Cucker22]). Because of this, a careful choice of how we measure errors is a fundamental step in the design and analysis of algorithms. A main example is numerical linear algebra, where it is commonplace to carefully choose a matrix norm depending on the problem at hand: the goal is to exploit the structure of the problem and optimise computational efficiency.

Unlike numerical linear algebra, a single norm – the Weyl norm – prevails in numerical algebraic geometry. The nice properties of the Weyl norm, ease of computing and unitary invariance, explain this prevalence. Nevertheless, the absence of complexity analyses using other norms in numerical algebraic geometry reflects badly on the theoretical strength of our analyses, which appear to rely on a specific choice of metric.

In this paper, we aim to show that using other norms is possible in numerical algebraic geometry. To do so, we consider an $L_\infty $ -norm in the space of polynomial systems and show how this leads to numerical algorithms and a complexity framework analogous to the one we have with the Weyl norm. Furthermore, we show that the change of norms leads to significant improvements in complexity bounds thanks to the better probabilistic behaviour of this $L_\infty $ norm with respect to the Weyl norm. We show this in three relevant cases: 1) computation of the homology of algebraic sets, 2) the Plantinga-Vegter algorithm and 3) the homotopy continuation method for quadratic polynomial systems.

We now discuss in more detail the aspects we have mentioned in passing to put our results in context within the wider setting of complexity theory for numerical algorithms and numerical algebraic geometry.

Complexity paradigm. The behaviour of numerical algorithms varies from input to input. This phenomenon is due not necessarily to the algorithms themselves but rather to the numerical sensitivity – how much the output varies with respect to a perturbation of the input – of the input we are processing. The numerical sensitivity of an input is captured by the so-called condition number. Then, in turn, condition numbers allow one to analyse numerical algorithms and explain why numerical algorithms handle some inputs faster than others.

Central to our paper is the fact that the choice of the metric under which we measure perturbations determines the condition number of the data. An example of this is given by the polynomial $X^d-1$ , which is well-conditioned (for the zero finding problem) with respect to the standard norm in equation (2.2) but badly conditioned with respect to the Weyl norm in equation (2.3) [Reference Bürgisser and Cucker14, Example 14.3].

A drawback of condition-based complexity analyses is that, as we don’t know a priori the condition of the input at hand, we cannot foresee the running time for this input. We can nonetheless get an idea of how the algorithm behaves in general by randomising the input. This allows one to obtain probabilistic estimates for the practical performance of the numerical algorithm.

Again, we note that the metric we choose to measure perturbations affects the probabilistic models we consider. This is so because probabilistic parameters such as the variance are always given with respect to some metric, so when we change the metric, we change the values of these parameters.

We refer to [Reference Bürgisser and Cucker14] for a more detailed overview of this complexity paradigm based on condition numbers. In the rest of the paper, we will show how this complexity framework works for each of the three cases mentioned above.

Choice of the norm. Arguably, one disadvantage of the $L_\infty $ -norm is that we don’t have an efficient way to approximate $\|~\|_{\infty }$ . For polynomials in $n+1$ homogeneous variables whose degrees are bounded by $\mathbf {D}$ , our current fastest algorithm takes time polynomial in $\mathbf {D}$ and exponential in n. However, the computation of $\|~\|_\infty $ amounts to a polynomial optimisation problem, and efficient algorithms exist for particular classes of polynomials. This is the case, for example, with sums of squares [Reference Laurent43, Reference Bhattiprolu, Guruswami and Lee10], sparse polynomials [Reference Dressler, Iliman and de Wolff31, Reference Chandrasekaran and Shah21] and other structures [Reference Barvinok5]. Unrestricted efficient algorithms are not expected to be designed because it is well-known that polynomial optimisation reduces to the feasibility problem over the reals, and the latter is ${\mathrm {NP}_{\mathbb {R}}}$ -complete. Nonetheless, for most applications we only need a coarse approximation of $\|~\|_{\infty }$ , which allows for some optimism.

Our choice of the $L_\infty $ -norm is due to the inequalities shown in Kellogg’s theorem (Theorem 2.13), which we haven’t found for other $L_p$ -norms. A way around Kellogg’s theorem for general $L_p$ -norms would certainly lead to new results regarding the use of these norms in algorithm analysis.

Despite the high cost of computing the $L_{\infty }$ norm, its use may yield substantially better cost bounds for some algorithms. This improvement rests on two facts:

  1. 1. For a homogeneous polynomial f with $n+1$ variables and degree $\mathbf {D}$ , we always have $ \left \lVert {f} \right \rVert _{\infty } \leq \left \lVert {f} \right \rVert _W$ , and for a random homogeneous polynomial $\mathfrak {f}$ , we have $ \left \lVert {\mathfrak {f}} \right \rVert _{\infty } \precsim \sqrt {n \log \mathbf {D}}$ , whereas $ \left \lVert {\mathfrak {f}} \right \rVert _W \sim \binom {n+\mathbf {D}}{n}^{\frac 12}$ . An analogous situation holds for polynomial systems (see Theorem 4.28 and Proposition 4.32).

  2. 2. Condition numbers with respect to the $L_\infty $ -norm yield condition-based complexity estimates (i.e., cost bounds in terms of both n, $\mathbf {D}$ and a condition number) almost identical to those obtained using the condition numbers with respect to the Weyl norm (see Section 3).

In this way, the reduction in the probabilistic estimates in passing to $ \left \lVert {~} \right \rVert _{\infty }$ from $ \left \lVert {~} \right \rVert _W$ immediately translates to reductions in the magnitude of the corresponding condition numbers and, in turn, reductions in the complexity estimates.

Considered algorithms. We showcase three algorithms where despite the high cost of computing the $L_{\infty }$ -norm, the reductions in the total cost bounds remain significant.

Firstly, in Section 4.1, we consider a family of algorithms (we refer to them as grid-based) that solve various problems in real algebraic and semialgebraic geometry. The best numerical algorithms for these problems have exponential complexity. In Section 4.1, we replace the Weyl norm by $\|~\|_{\infty }$ in the design of one such algorithm (to compute Betti numbers); and in Section 4.3, we show a decrease in its cost bounds. We take advantage of the fact that there is only one norm computation, and it is done, so to speak, along the way. The gain in the reduction of the estimate for the number of iterations directly yields a reduction in the total cost bound (see Corollary 4.31).

Secondly, in Section 4.2, we consider the Plantinga-Vegter algorithm as it is described and analysed in [Reference Cucker, Ergür and Tonelli-Cueto23]. Again, we replace the Weyl norm by $\|~\|_{\infty }$ in the algorithm’s design results in improved cost bounds. And again, the computation of $\|~\|_\infty $ is not a burden as it is done only once, and its cost is dominated by that of the rest of the algorithm. The Plantinga-Vegter algorithm is usually considered with $n=2$ or $n=3$ . Remark 4.35 exhibits the improvement achieved on average complexity bounds for these two cases. For larger values of n, the improvement is more substantial.

Thirdly, in Section 5, we consider the problem of computing a zero of a system of complex quadratic equations. For this question, a particular case of Smale’s 17th problem, we consider the algorithms proposed in [Reference Beltrán and Pardo9, Reference Bürgisser and Cucker13] and, again, design versions of them where the Weyl norm is replaced by $\|~\|_{\infty }$ . Again, this results in a small but measurable reduction in the cost bounds (from $n^7$ to $n^{6.875}$ ). A crucial fact in achieving this is that even though n is general, we can find an efficient way to compute $\|~\|_{\infty }$ using the fact that $\mathbf {D}=2$ .

In all three cases, we are able to show that the use of $L_{\infty }$ -norm yields a clear reduction in the estimates for the expected number of iterations. We believe this is a common pattern. But in general, the reduction in the number of steps does not immediately translate into a reduction in total computational cost. This motivates the search for efficient algorithms that (roughly) approximate $\|~\|_{\infty }$ and for a better understanding of the complexity and accuracy of computing with $L_p$ -norms in polynomial spaces.

Organisation of the paper. In Section 2, we define the norms that will be considered in this paper and work out several examples. We also recall basic properties of these norms and highlight their differences from the Weyl norm. Then, in Section 3, we define condition numbers $\mathsf {M}$ and $\mathsf {K}$ that scale with the $L_{\infty }$ -norm. These condition numbers are similar to their widely used Weyl versions $\mu _{\mathrm {norm}}$ and $\kappa $ (for complex and real problems, respectively). We also prove in Section 3 that the main properties of $\mu _{\mathrm {norm}}$ and $\kappa $ – those allowing them to feature in condition-based cost estimates – hold for $\mathsf {M}$ and $\mathsf {K}$ . Section 4.1, Section 4.2 and Section 5 are the home of three algorithms that are designed using $L_{\infty }$ -scaled condition numbers. We compare the cost bounds of these algorithms to those of their Weyl counterparts and highlight computational gains.

We conclude in Appendix A with a minor digression. Because a natural habitat for functional norms is spaces of continuous functions, we consider extensions of the real condition number $\kappa $ to the space $C^1[q]:=C^1(\mathbb {S}^n,\mathbb {R}^q)$ , and we prove (somehow unexpectedly) Condition Number Theorems for these extensions. We do not analyse algorithms here. We nonetheless point out that substantial literature on algorithms on spaces of continuous functions exists [Reference Traub, Wasilkowski, Woźniakowski, Werschulz and Boult57, Reference Plaskota50, Reference Novak, Sloan, Traub and Woźniakowski48], where these theorems might be useful.

2 Norms for polynomials

Let $\mathbb {F}$ be either $\mathbb {R}$ or $\mathbb {C}$ . Let also $n,d\in \mathbb {N}$ , $n,d\ge 1$ . We denote by $\mathcal {H}^{\mathbb {F}}_d[1]$ the linear space of homogeneous polynomials of degree d in the $n+1$ variables $X_0,X_1,\ldots ,X_n$ with coefficients in $\mathbb {F}$ . Let $\boldsymbol {d}=(d_1,\ldots ,d_q)\in \mathbb {N}^q$ and $n\in \mathbb {N}$ as above. We denote by $\mathcal {H}_{\boldsymbol {d}}^{\mathbb {F}}[q]$ the space $\mathcal {H}^{\mathbb {F}}_{d_1}[1]\times \cdots \times \mathcal {H}^{\mathbb {F}}_{d_q}[1]$ . If $\mathbb {F}$ is clear from the context, or if it is not relevant to the argument, we will omit the superscript. We will use the following conventions for dimension counting:

$$ \begin{align*}N_i:=\binom{n+d_i}{d_i}=\dim_{\mathbb{F}}\mathcal{H}^{\mathbb{F}}_{d_i}[1] \quad \text{and}\quad N:=\sum_{i=1}^q\binom{n+d_i}{d_i}=\dim_{\mathbb{F}} \mathcal{H}_{\boldsymbol{d}}^{\mathbb{F}}[q].\end{align*} $$

We also use $\mathbf {D}:=\max \{d_1,\ldots ,d_q\}$ and denote by $\Delta $ the $q\times q$ diagonal matrix with $d_i$ in its ith diagonal entry.

In all that follows, $\mathbb {S}^n:=\{x\in \mathbb {R}^{n+1}\mid \|x\|_{2}=1\}$ will be the (real) n-sphere and $\mathbb {P}^n:=\mathbb {C}^{n+1}/\mathbb {C}^*$ the complex projective space of dimension n. We note that there will be no ambiguity, as the sphere is the usual space to work with real polynomials and the projective space is the usual one for complex polynomials.

Remark 2.1. In what follows, we will write $z\in \mathbb {P}^n$ instead of $[z]\in \mathbb {P}^n$ , and we will assume that the representative $z\in \mathbb {C}^{n+1}$ always satisfies $\|z\|_{2}=1$ . This simplifies the form of many of our definitions. This convention can be made without loss of generality as every point in $\mathbb {P}^n$ has a representative of norm $1$ .

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:

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

Sketch of proof.

It is a direct consequence of the inequalities between p-means. $\Box $

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

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

Sketch of proof.

We only need to show this in the case $q=1$ . Now both the Weyl norm and the complex $L_2$ -norm are unitarily invariant Hermitian norms of $\mathcal {H}_d^{\mathbb {C}}$ . For the Weyl norm, see [Reference Bürgisser and Cucker14, Theorem 16.3]; for the complex $L_2$ -norm, this is property (I). Since $\mathcal {H}_d^{\mathbb {C}}$ is an irreducible representation of $\mathscr {U}(n+1)$ , this means the two norms are equal up to a constant. Using Example 2.4 with $f=X_0^d$ , one can check that this constant is $\sqrt {N}$ . $\Box $

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 $p\geq 2$ ,

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

Sketch of proof.

By Proposition 2.2, $f\mapsto f(x)$ is an orthogonal projection with respect to the Weyl norm, so $\|f(x)\|_2\leq \|f\|_W$ . Hence, for every $x \in S^{n-1}$ , $\|f(x)\|_p \leq \|f(x)\|_2\leq \|f\|_W$ , where the first inequality follows from Minkowski’s inequality. $\Box $

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

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

Proof. The first part follows from Proposition 2.9 and 2.10. The left-hand side of the second part uses Proposition 2.10.

Now, for $f\in \mathcal {H}_d[1]$ , Corollary 2.20 implies that for each $\alpha $ , $|f_\alpha |=\left \|\frac {1}{\alpha !}\overline {\mathrm {D}}_xf\right \|\leq \binom {d}{\alpha }$ . The right-hand inequality follows from here.

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?

Remark 2.18. Examples 2.16 and 2.17 motivate the search of a randomised Kellogg’s theorem that holds with high probability for random polynomials and has a tighter right-hand side.

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.

Remark 2.19. We note that the application of [Reference Kellogg40, Theorem IV] using the scaling with the diagonal matrix was not used in [Reference Ergür, Paouris and Rojas33, Theorem 2.4] and [Reference Ergür, Paouris and Rojas34]. This can be used to improve by a factor of the degree some of the bounds there.

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

Proof. It follows from Theorem 2.13 by induction, followed by an application of Corollary 2.14.

Remark 2.21. Although the results in this section were proved only for $\|~\|^{\mathbb {F}}_{\infty }$ , some of them can be generalised to other norms. For example, similar results can be obtained for $\|~\|^{\mathbb {R}}_2$ (see [Reference Seeley52]) and certainly for other norms. We defer to future work the application of these extensions to the analysis of numerical algorithms in algebraic geometry. We also note that Corollary 2.14 for $\mathbb {F}=\mathbb {R}$ can be generalised to smooth real algebraic varieties other than the sphere (see [Reference Bos, Levenberg, Milman and Taylor11]).

3 Condition numbers for the $L_\infty $ -norm

In this section, we will consider condition numbers that capture ‘how near to being singular’ a system $f\in \mathcal {H}_{\boldsymbol {d}}[q]$ is at a point $x\in \mathbb {S}^n$ . We will define condition numbers and develop a geometric understanding of them for the $L_\infty $ -norms defined in the preceding section.

Recall the local and global versions of the real condition number $\kappa $ used in [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick, Malajovich and Wschebor26, Reference Cucker, Krick, Malajovich and Wschebor27, Reference Cucker, Krick and Shub28]. For $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $x\in \mathbb {S}^n$ , they are defined by

(3.1) $$ \begin{align} \kappa(f,x):=\frac{\|f\|_W}{\sqrt{\|f(x)\|_2^2+\left\|\mathrm{D}_xf^\dagger\Delta^{1/2}\right\|^{-2}_{2,2}}}~\text{ and }~\kappa(f):=\sup_{y\in\mathbb{S}^n}\kappa(f,y). \end{align} $$

Here, for a surjective linear map A, $A^\dagger :=A^*(AA^*)^{-1}$ denotes its Moore-Penrose inverse [Reference Bürgisser and Cucker14, Section 1.6]. Also recall the $\mu $ -condition number introduced by Shub and Smale [Reference Shub and Smale53]: for $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ and $\zeta \in \mathbb {P}^n$ , $\mu (f,\zeta )$ is defined by

(3.2) $$ \begin{align} \mu_{\mathrm{norm}}(f,\zeta):= \|f\|_W\left\|\mathrm{D}_{\zeta} f^\dagger\Delta^{1/2}\right\|_{2,2}. \end{align} $$

Remark 3.1. By convention, we assume that $\|A^\dagger \|_{2,2}=\infty $ when A is not surjective. We do this because for $A\in \mathbb {C}^{q\times n}$ surjective,

$$ \begin{align*} \left\|A^\dagger\right\|^{-1}_{2,2}=\sigma_q(A), \end{align*} $$

where $\sigma _q$ is the qth singular value. As the latter is continuous, this choice guarantees that $A\mapsto \|A^\dagger \|_{2,2}^{-1}$ is continuous.

Following these ideas, we define the real local condition number of $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ at $x\in \mathbb {S}^n$ as

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

and the real global condition number of $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ as

(3.4) $$ \begin{align} \mathsf{K}(f):=\sup_{y\in\mathbb{S}^n}\mathsf{K}(f,y). \end{align} $$

And we define the complex local condition number of $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ at $\zeta \in \mathbb {P}^n$ as

(3.5) $$ \begin{align} \mathsf{M}(f,\zeta)=\sqrt{q}\|f\|_{\infty}^{\mathbb{C}}\left\|\mathrm{D}_{\zeta}f^\dagger\Delta\right\|_{2,2} \end{align} $$

and the complex global condition number of $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ (with $q\le n$ ) as

(3.6) $$ \begin{align} \mathsf{M}(f):=\sup\{\mathsf{M}(f,\zeta)\mid \zeta\in\mathbb{P}^n,\,f(\zeta)=0\}. \end{align} $$

We can see that $\mathsf {K}$ is a variant of $\kappa $ and $\mathsf {M}$ is a variant of $\mu _{\mathrm {norm}}$ . We note that the main difference lies in the fact that we are substituting all occurrences of $\|~\|_W$ with occurrences of $\|~\|_\infty $ . The fact that we use a different scaling factor ( $\Delta ^{1/2}$ instead of $\Delta $ ) or different norms for vectors ( $\|~\|_\infty $ instead of $\|~\|_2$ and so on) only affects these quantities up to a $\sqrt {2q\mathbf {D}}$ factor. This has little consequence for complexity. We will be more explicit in Proposition 4.27. Note that despite these changes, we still have that the local condition numbers, $\mathsf {K}$ and $\mathsf {M}$ , become $\infty $ at a singular zero and that they are finite otherwise.

The remainder of this section is devoted to proving the main properties of $\mathsf {K}$ and $\mathsf {M}$ , which are the reason we defined these numbers the way we did. The properties we will show are those needed for a condition-based complexity analyses of the algorithms in Sections 4 and 5 following the lines of the analyses in [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick and Shub28, Reference Bürgisser, Cucker and Lairez15, Reference Bürgisser, Cucker and Tonelli-Cueto16, Reference Bürgisser, Cucker and Tonelli-Cueto17] (see also [Reference Tonelli-Cueto55]) and [Reference Bürgisser and Cucker14, Chapter 17].

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.

Theorem 3.2. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {R}}[q]$ and $x\in \mathbb {S}^n$ . The following holds:

  • Regularity inequality: Either

    $$ \begin{align*}\frac{\|f(x)\|}{\sqrt{q}\|f\|_\infty^{\mathbb{R}}}\geq\frac{1}{\mathsf{K}(f,x)}\text{ or }\sqrt{q}\|f\|_\infty^{\mathbb{R}}\left\|\mathrm{D}_xf^\dagger\Delta\right\|_{2,2}\leq\mathsf{K}(f,x).\end{align*} $$
    In particular, if $\mathsf {K}(f,x)\frac {\|f(x)\|}{\sqrt {q}\|f\|_\infty ^{\mathbb {R}}}<1$ , then $\mathrm {D}_xf:\mathrm {T}_x\mathbb {S}^n\rightarrow \mathbb {R}^q$ is surjective and its pseudoinverse $(\mathrm {D}_xf)^{\dagger }$ exists.
  • 1st Lipschitz property: The maps

    $$ \begin{align*} \begin{array}{rl}\mathcal{H}_{\boldsymbol{d}}^{\mathbb{R}}[q]&\rightarrow [0,\infty)\\ g&\mapsto \dfrac{\|g\|_\infty^{\mathbb{R}}}{\mathsf{K}(g,x)}\end{array}\qquad\qquad\textrm{and}\qquad\qquad\begin{array}{rl}\mathcal{H}_{\boldsymbol{d}}^{\mathbb{R}}[q]&\rightarrow [0,\infty)\\ g&\mapsto \dfrac{\|g\|_\infty^{\mathbb{R}}}{\mathsf{K}(g)}\end{array} \end{align*} $$
    are $1$ -Lipschitz with respect to the real $L_\infty $ -norm. In particular,
    $$ \begin{align*}\mathsf{K}(f,x)\geq 1~\text{ and }~\mathsf{K}(f)\geq 1.\end{align*} $$
  • 2nd Lipschitz property: The map

    $$ \begin{align*} \mathbb{S}^n&\rightarrow [0,1]\\ y&\mapsto \frac{1}{\mathsf{K}(f,y)} \end{align*} $$
    is $\mathbf {D}$ -Lipschitz with respect to the geodesic distance on $\mathbb {S}^n$ .
  • Higher derivative estimate: If $\mathsf {K}(f,x)\frac {|f(x)|}{\|f\|_\infty ^{\mathbb {R}}}<1$ , then

    $$ \begin{align*} \gamma(f,x)\leq \frac{1}{2}(\mathbf{D}-1)\mathsf{K}(f,x). \end{align*} $$

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]).

Theorem 3.3. Let $f\in \mathcal {H}_{\boldsymbol {d}}[n]$ and $\zeta \in \mathbb {S}^n$ such that $f(\zeta )=0$ . Let $z\in \mathbb {S}^n$ be such that $\mathrm {dist}_{\mathbb {S}}(z,\zeta )\leq \frac {1}{45}$ and $\mathrm {dist}_{\mathbb {S}}(z,\zeta )\gamma (f,\zeta )\le 0.17708$ . Then z is an approximate zero of f with associated zero $\zeta $ .

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).

Theorem 3.5. Let $f\in \mathcal {H}_{\boldsymbol {d}}^{\mathbb {C}}[q]$ and $\zeta \in \mathbb {P}^n$ . The following holds:

  • 1st Lipschitz property: The maps

    $$ \begin{align*} \begin{array}{rl}\mathcal{H}_{\boldsymbol{d}}^{\mathbb{C}}[q]&\rightarrow [0,\infty)\\ g&\mapsto \frac{\|g\|_\infty^{\mathbb{C}}}{\mathsf{M}(g,\zeta)}\end{array}\qquad\qquad\textrm{and}\qquad\qquad\begin{array}{rl}\mathcal{H}_{\boldsymbol{d}}^{\mathbb{C}}[q]&\rightarrow [0,\infty)\\ g&\mapsto \frac{\|g\|_\infty^{\mathbb{C}}}{\mathsf{M}(g)}\end{array} \end{align*} $$
    are $1$ -Lipschitz with respect to the complex $L_\infty $ -norm. In particular,
    $$ \begin{align*}\mathsf{M}(f,\zeta)\geq 1~\text{ and }~\mathsf{M}(f)\geq 1.\end{align*} $$
  • 2nd Lipschitz property: The map

    $$ \begin{align*} \mathbb{P}^n&\rightarrow [0,1]\\ \eta&\mapsto \frac{1}{\mathsf{M}(f,\eta)} \end{align*} $$
    is $\mathbf {D}$ -Lipschitz with respect to the geodesic distance $\mathrm {dist}_{\mathbb {P}}$ on $\mathbb {P}^n$ .
  • Higher derivative estimate: We have

    $$ \begin{align*} \gamma(f,\zeta)\leq \frac{1}{2}(\mathbf{D}-1)\mathsf{M}(f,\zeta). \end{align*} $$

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 Numerical algorithms in real algebraic geometry

There is a growing literature on numerical algorithms that addresses basic computational tasks in real algebraic geometry, such as counting real zeros [Reference Cucker, Krick, Malajovich and Wschebor25, Reference Cucker, Krick, Malajovich and Wschebor26, Reference Cucker, Krick, Malajovich and Wschebor27], computing homology of algebraic [Reference Cucker, Krick and Shub28] and semialgebraic sets [Reference Bürgisser, Cucker and Lairez15, Reference Bürgisser, Cucker and Tonelli-Cueto16, Reference Bürgisser, Cucker and Tonelli-Cueto17], and meshing real curves and surfaces [Reference Plantinga and Vegter49, Reference Cucker, Ergür and Tonelli-Cueto23]. These works rely on condition numbers to control precision and estimate computational complexity.

In this section, we show how the complexity estimates in these works are improved by using the real $L_\infty $ -norm in the algorithm’s design. These improvements rely on three observations:

  1. 1. The only properties of the real condition number $\kappa $ that are used in the complexity analyses are those stated in Theorem 3.2: the regularity inequality, the 1st and 2nd Lipschitz properties and the higher derivative estimate. As these properties hold as well for $\mathsf {K}$ , an almost identical condition-based cost analysis can be derived when we pass from the Weyl norm to the real $L_\infty $ -norm and from $\kappa $ to $\mathsf {K}$ . We showcase this in Section 4.1 and Section 4.2.

  2. 2. When we consider random input models, the gains in the complexity estimates become more evident. In Section 4.3, we show that the ratio of the new $\mathsf {K}$ to $\kappa $ is typically of the order of $\sqrt {n}/\sqrt {N}$ for a random polynomial system. Since $N \sim n^d$ for $n>d$ and $N \sim d^n$ for $d>n$ , this yields a significant reduction in the complexity estimates.

  3. 3. Computing the Weyl norm is cheaper than computing the real $L_\infty $ -norm, but this does not affect the overall complexity: We only compute the $L_\infty $ -norm once, and the cost of this computation is dominated by that of the remaining steps.

In what follows, we will focus on algorithms dealing with real algebraic sets. The algorithms we have in mind are the ones 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 the Plantinga-Vegter algorithm [Reference Plantinga and Vegter49] as described and analysed in [Reference Cucker, Ergür and Tonelli-Cueto24] (compare to [Reference Cucker, Ergür and Tonelli-Cueto23]). Our condition number $\mathsf {K}$ as defined in the preceding section will improve the overall computational complexity of these algorithms. Similar results can be obtained for the algorithms dealing with semialgebraic sets in [Reference Bürgisser, Cucker and Lairez15, Reference Bürgisser, Cucker and Tonelli-Cueto16, Reference Bürgisser, Cucker and Tonelli-Cueto17] (compare to [Reference Tonelli-Cueto55]) using natural extensions $\overline {\mathsf {K}}$ and $\mathsf {K}_*$ of the condition numbers $\overline {\kappa }$ and $\kappa _*$ used in these papers.

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 $\mathcal {O}\left (2^{n\log n+nk}\right )$ .

Remark 4.2. The grid construction in Proposition 4.1, which occurs in [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 not optimal. This is due to the $2^{n\log n}$ factor in the estimates, which can be decreased to $2^{\mathcal {O}(n)}$ . An algorithm doing this – that is, constructing a spherical $2^{-k}$ -net of size $2^{\mathcal {O}(n)}2^{k(n+1)}$ in $2^{\mathcal {O}(n)}2^{k(n+1)}$ -time – is given in [Reference Alon, Lee, Shraibman and Vempala2, Theorem 1.9(1)]. We use the suboptimal result of Proposition 4.1 to focus on the effect of just changing the norm when comparing the old and new versions of the algorithms. But we observe here that by using the nets in [Reference Alon, Lee, Shraibman and Vempala2], one can remove the $\log (n)$ factors in the exponents.

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 $K\rho =2/\sqrt {\pi }$ <