2.1 The classical setting

Numerical continuation algorithms have been so far the main tool for the complexity analysis of numerical solving of polynomial systems. We present, here, the main line of the theory as developed by [Reference Beltrán3, Reference Beltrán and Pardo7, Reference Beltrán and Pardo8, Reference Shub and Smale40, Reference Shub and Smale41, Reference Shub and Smale43, Reference Shub and Smale44]. The general idea to solve a polynomial system $F \in \mathcal {H}$ consists of embedding F in a one-parameter continuous family $(F_t)_{t\in [0,1]}$ of polynomial systems, such that $F_1 = F$ and a zero of $F_0$ , say, $\zeta _0\in \mathbb {P}^n$ , is known. Then, starting from $t=0$ and $z=\zeta _0$ , t and z are updated to track a zero of $F_t$ all along the path from $F_0$ to $F_1$ , as follows:

where $\Delta t$ needs to be defined. The idea is that z always stays close to $\zeta _t$ , the zero of $F_t$ obtained by continuing $\zeta _0$ . To ensure correctness, the increment $\Delta t$ should be chosen small enough. But the bigger $\Delta t$ is, the fewer iterations will be necessary, meaning a better complexity. The size of $\Delta t$ is typically controlled with, on the one hand, effective bounds on the variations of the zeros of $F_t$ as t changes, and on the other hand, effective bounds on the convergence of Newton’s iteration. The general principle to determine $\Delta t$ is the following, in very rough terms because a precise argument generally involves lengthy computations. The increment $\Delta t$ should be small enough so that $\zeta _{t}$ is in the basin of attraction around $\zeta _{t+\Delta t}$ of Newton’s iteration for $F_{t+\Delta t}$ . This leads to the rule-of-thumb $\| \Delta \zeta _t \| \rho (F_{t+\Delta t}, \zeta _{t+\Delta t}) \lesssim 1$ , where $\Delta \zeta _t = \zeta _{t+\Delta t} - \zeta _{t}$ and $\rho (F_t, \zeta _t)$ is the inverse of the radius of the basin of attraction of Newton’s iteration. A condition that we can rewrite as

(2.1) $$ \begin{align} \frac{1}{\Delta t} \gtrsim \rho(F_t, \zeta_t) \left\| \frac{\Delta \zeta_t}{\Delta t} \right\|,\\[-15pt]\nonumber \end{align} $$

assuming that

(2.2) $$ \begin{align} \rho(F_{t+\Delta t}, \zeta_{t+\Delta t}) \simeq \rho(F_t, \zeta_t).\\[-15pt]\nonumber \end{align} $$

The factor $\frac {\Delta \zeta _t}{\Delta t}$ is almost the derivative $\dot \zeta _t$ of $\zeta _t$ with respect to t. It is generally bounded using a condition number $\mu (F_t, \zeta _t)$ , that is the largest variation of the zero $\zeta _t$ after a perturbation of $F_t$ in $\mathcal {H}$ , so that

(2.3) $$ \begin{align} \left\| \frac{\Delta \zeta_t}{\Delta t} \right\| \simeq \| \dot \zeta_t \| \leqslant \mu(F_t, \zeta_t) \|\dot F_t\|,\\[-15pt]\nonumber \end{align} $$

where $\dot F_t$ (respectively, $\dot \zeta _t$ ) is the derivative of F (respectively, $\zeta _t$ ) with respect to t, and the right-hand side is effectively computable. The parameter $\rho (F_t, \zeta _t)$ is much deeper. Smale’s $\alpha $ -theory has been a preferred tool to deal with it in many complexity analyses. The number $\gamma $ takes a prominent role in the theory and controls the convergence of Newton’s iteration [Reference Smale45]: $\rho (F_t, \zeta _t) \lesssim \gamma (F_t, \zeta _t)$ (for the definition of $\gamma (F,\zeta )$ , e.g. see Equation (8) in Part I.) So we obtain the condition

(2.4) $$ \begin{align} \frac{1}{\Delta t} \gtrsim \gamma(F_t, \zeta_t) \mu(F_t, \zeta_t) \|\dot F_t\| \\[-15pt]\nonumber\end{align} $$

that ensures the correctness of the algorithm. A rigorous argument requires a nice behaviour of both factors $\gamma (F_t, \zeta _t)$ and $\mu (F_t, \zeta _t)$ as t varies, this is a crucial point, especially in view of the assumption (2.2). The factor $\|\dot F_t\|$ is generally harmless; the factor $\mu (F_t, \zeta _t)$ is important, but the variations with respect to t are generally easy to handle; however, the variations of $\gamma (F_t, \zeta _t)$ are more delicate. This led [Reference Shub and Smale40] to consider the upper bound (called ‘higher-derivative estimate’)

(2.5) $$ \begin{align} \gamma(F, z) \lesssim \mu(F, z),\\[-15pt]\nonumber \end{align} $$

with the same $\mu $ as above, and the subsequent correctness condition

(2.6) $$ \begin{align} \frac{1}{\Delta t} \gtrsim \mu(F_t, \zeta_t)^2 \|\dot F_t\|.\\[-15pt]\nonumber \end{align} $$

Choosing at each iteration $\Delta t$ to be the largest possible value allowed by (2.6), we obtain a numerical continuation algorithm, with adaptive step length, whose number K of iterations is bounded, as shown first by [Reference Shub39], by

(2.7) $$ \begin{align} K \lesssim \int_0^1 \mu(F_t, \zeta_t)^2 \|\dot F_t\| \mathrm{d} t. \end{align} $$

It remains to choose the starting system $F_0$ , with a built-in zero $\zeta _0$ , and the path from $F_0$ to $F_1$ . For complexity analyses, the most common choice of path is a straight-line segment in the whole space of polynomial systems $\mathcal {H}$ . For the choice of the starting system $F_0$ , [Reference Beltrán and Pardo6, Reference Beltrán and Pardo7] have shown that a Kostlan random system is a relevant choice and that there is a simple algorithm to sample a random system with a known zero. If $F_1$ is also a random Gaussian system, then all the intermediate systems $F_t$ are also random Gaussian, and using (2.7), we obtain a bound, following Beltrán and Pardo, on the expected number of iterations in the numerical continuation from $F_0$ to $F_1$ :

(2.8) $$ \begin{align} \mathbb{E}_{F_0, \zeta_0, F_1}[K] \simeq \mathbb{E}_{F, \zeta}[\mu(F, \zeta)^2] \simeq \dim \mathcal{H}, \end{align} $$

where $\zeta $ is a random zero of F. The dimension of $\mathcal {H}$ is the number of coefficients in F, it is the input size. For n equations of degree ${D}$ in n variables, we compute

(2.9) $$ \begin{align} \dim \mathcal{H} = n \binom{n+{D}}{n}. \end{align} $$

This is larger than any polynomial in n and ${D}$ (as n and ${D}$ go to $\infty $ ), but is much smaller than ${D}^n$ , the generic number of solutions of such a system. The cost of an iteration (computing the step size and performing one Newton iteration) is also bounded by the input size. So we have an algorithm whose average complexity is polynomial in the input size. This is a major complexity result because it breaks the $\operatorname {poly}({D}^n)$ barrier set by algorithms that compute all solutions simultaneously. However, the bound (2.8) on the expected number of iterations is still much larger than what heuristic algorithms seem to achieve.

A first idea to design a faster algorithm would be to search for a better continuation path in order to lower the right-hand side in (2.7). Such paths do exist and can give a $\operatorname {poly}(n,{D})$ bound on $\mathbb {E}[K]$ [Reference Beltrán and Shub9]. Unfortunately, their computation requires, in the current state of the art, to solve the target system first. A second approach focuses on sharpening the correctness condition (2.6), that is, on making bigger continuation steps. The comparison of (2.6) with heuristics shows that there is room for improvement [Reference Beltrán and Leykin4, Reference Beltrán and Leykin5]. In devising this condition, two inequalities are too generous. Firstly, Inequality (2.3) bounds the variation of $\zeta _t$ by the worst-case variation. The average worst-case variation can only grow with the dimension of the parameter space, $\dim \mathcal {H}$ , and it turns out to be much bigger than the average value of $\|\dot \zeta _t\|$ , which is $\operatorname {poly}(n,{D})$ . This was successfully exploited by [Reference Armentano, Beltrán, Bürgisser, Cucker and Shub1] to obtain the bound $\mathbb {E}[K] \lesssim \sqrt {\dim \mathcal {H}}$ for random Gaussian systems. They used straight-line continuation paths but a finer computation of the step size. The other inequality that turns out to be too coarse is (2.5): the higher derivatives need to be handled more accurately.

2.2 Rigid continuation paths

In Part I, we introduced rigid continuation paths to obtain, in the case of random Gaussian systems, the bound $\mathbb {E}[K] \leqslant \operatorname {poly}(n, {D})$ . To solve a polynomial system $F = (f_1,\dotsc ,f_n) \in \mathcal {H}$ in $n+1$ homogeneous variables, we consider continuation paths having the form

(2.10) $$ \begin{align} F_t \,{\stackrel{.}{=}}\, \left( f_1 \circ u_1^{-1}(t),\dotsc, f_n\circ u_n^{-1}(t) \right), \end{align} $$

where $u_1(t),\dotsc ,u_n(t) \in U(n+1)$ are unitary matrices depending on the parameter t, with $u_i(T)=\operatorname {id}$ , for some $T> 0$ . We could choose $T = 1$ , but instead, we impose $\| \dot u_1 \|^2 + \dotsb + \|\dot u_n\|^2 \leqslant 1$ (the path is called 1-Lipschitz continuous), following Part I. Note that we can choose $T\leqslant 4n$ (see I, Section 4.3.2). The parameter space for the numerical continuation is not $\mathcal {H}$ anymore but $U(n+1)^n$ , denoted ${\mathcal {U}}$ , a real manifold of dimension $n^3$ . For ${\mathbf u} = (u_1,\dotsc ,u_n) \in {\mathcal {U}}$ and $F \in \mathcal {H}$ , we denote

(2.11) $$ \begin{align} {\mathbf u} \cdot F \,{\stackrel{.}{=}}\, \left( f_1 \circ u_1^{-1},\dotsc, f_n\circ u_n^{-1} \right) \in \mathcal{H}. \end{align} $$

We developed in this setting an analogue of Beltrán and Pardo’s algorithm. Firstly, we sample uniformly ${\mathbf v} \in {\mathcal {U}}$ together with a zero of the polynomial system ${\mathbf v} \cdot F$ . The same kind of construction as in the Gaussian case makes it possible to perform this operation without solving any polynomial system (only n univariate equations). Then, we construct a 1-Lipschitz-continuous path $({\mathbf u}_t)_{t\in [0,T]}$ in ${\mathcal {U}}$ between ${\mathbf v}$ and the unit $\mathbf {1}_{\mathcal {U}}$ in ${\mathcal {U}}$ , and perform numerical continuation using $F_t\, {\stackrel {.}{=}}\, {\mathbf u}_t \cdot F$ . The general strategy sketched in Section 2.1 applies, but the rigid setting features important particularities. The most salient of them is the average conditioning, that is, the average worst-case variation of $\zeta _t$ with respect to infinitesimal variations of ${\mathbf u}_t$ . It is now $\operatorname {poly}(n)$ (see I, Section 3.2), mostly because the dimension of the parameter space is $\operatorname {poly}(n)$ . Besides, the way the continuation path is designed preserves the geometry of the equations. This is reflected in a better behaviour of $\gamma (F_t, \zeta _t)$ as t varies, which makes it possible to use an upper bound much finer than (2.5), that we called the split $\gamma $ number. In the case of a random Gaussian input, we obtained in the end a $\operatorname {poly}(n,{D})$ bound on the average number of iterations for performing numerical continuation along rigid paths.

2.3 The split $\gamma $ number

Computing a good upper bound of the $\gamma $ number is the key to make bigger continuation steps. We recall, here, the upper bound introduced in Part I. The incidence condition number of $F = (f_1,\dotsc ,f_n)$ at z is

(2.12)

where $\dagger $ denotes the Moore–Penrose pseudoinverse [Reference Bürgisser and Cucker15, p. 17] and $F_z$ the normalised system

(2.13) $$ \begin{align} F_z\, {\stackrel{.}{=}}\, \left( \frac{f_1}{\|\mathrm{d} _z f_1\|},\dotsc, \frac{f_n}{\|\mathrm{d} _z f_n\|} \right). \end{align} $$

When z is a zero of F, this quantity depends only on the angles formed by the tangent spaces at z of the n hypersurfaces $\left \{f_i=0 \right \}$ (see I, Sections 2.1 and 3 for more details). It is closely related to the intersection condition number introduced by [Reference Bürgisser12]. In the context of rigid paths, it is also the natural condition number: the variation of a zero $\zeta $ of a polynomial system ${\mathbf u} \cdot F$ under a perturbation of ${\mathbf u}$ is bounded by $\kappa ({\mathbf u} \cdot F,\zeta )$ (Lemma I.16). Moreover, F being fixed, if ${\mathbf u} \in {\mathcal {U}}$ is uniformly distributed and if $\zeta $ is a uniformly distributed zero of ${\mathbf u} \cdot F$ , then $\mathbb {E}[\kappa ({\mathbf u}\cdot F, \zeta )^2] \leqslant 6n^2$ (Proposition I.17).

The split $\gamma $ number (see I, Section 2.5 for more details) is defined as

(2.14) $$ \begin{align} \hat{\gamma}(F, z) \,{\stackrel{.}{=}} \,\kappa(F, z) \left( \gamma(f_1,z)^2 + \dotsb + \gamma(f_n, z)^2 \right)^{\frac12}. \end{align} $$

It tightly upper bounds $\gamma (F, z)$ in that (Theorem I.13)

(2.15) $$ \begin{align} \gamma(F, z) \leqslant \hat \gamma(F, z) \leqslant n \kappa(F, z) \gamma(F, z). \end{align} $$

Whereas $\gamma (F, z)$ does not behave nicely as a function of F, the split variant behaves well in the rigid setting: F being fixed, the function ${\mathcal {U}} \times \mathbb {P}^n \to \mathbb {R}$ , $({\mathbf u}, z) \mapsto \hat {\gamma }({\mathbf u} \cdot F, z)^{-1}$ is 13-Lipschitz continuous (Lemma I.21).Footnote ⁶ This makes it possible to perform numerical continuation. Note that we need not compute $\gamma $ exactly, an estimate within a fixed ratio is enough. For computational purposes, we rather use the variant $\gamma _{\mathrm {Frob}}$ , defined in (1.3), in which the operator norm is replaced by a Hermitian norm. It induces a split $\gamma _{\mathrm {Frob}}$ number

(2.16) $$ \begin{align} \hat{\gamma}_{\mathrm{Frob}}(F, z)\, {\stackrel{.}{=}} \,\kappa(F, z) \left( \gamma_{\mathrm{Frob}}(f_1,z)^2 + \dotsb + \gamma_{\mathrm{Frob}}(f_n, z)^2 \right)^{\frac12} \end{align} $$

as in (2.14). Algorithm 1 describes the computation of an approximate zero of a polynomial system ${\mathbf u} \cdot F$ , given a zero of some ${\mathbf v}\cdot F$ (it is the same as Algorithm I.2, with $\hat {\gamma }_{\mathrm {Frob}}$ for g and $C=15$ , which gives the constant 240 that appears in Algorithm 1). As an analogue of (2.7), Theorem I.23 bounds the number K of continuation steps performed by Algorithm 1 as an integral over the continuation path:

(2.17) $$ \begin{align} K \leqslant 325 \int_0^T \kappa({\mathbf w}_t\cdot F, \zeta_t) \hat \gamma_{\mathrm{Frob}}({\mathbf w}_t\cdot F, \zeta_t) \mathrm{d} t. \end{align} $$

Based on this bound, we obtained in Part I the following average analysis.

Let $F = (f_1,\dotsc ,f_n) \in {\mathcal {H}}$ be a square-free polynomial system. Then, for uniformly random ${\mathbf u},{\mathbf v} \in {\mathcal {U}}$ and a uniformly random zero $\zeta \in {{\mathbb {P}}^n}$ of ${\mathbf v}\cdot F$ , Algorithm 1 terminates almost surely and outputs an approximate zero of ${\mathbf u}\cdot F$ . Moreover, the number K of continuation steps it performs satisfies ${\mathbb {E}}[K] \leqslant 9000 n^3\,\Gamma (F)$ (Theorem I.27 with $\mathfrak g_i = {\gamma _{\mathrm {Frob}}}$ and $C'=5$ —by Lemma I.31—which multiplies the constant 1,800 in its statement). Here, $\Gamma (F)$ denotes the crucial parameter introduced in (1.6).

In case we cannot compute $\gamma _{\mathrm {Frob}}$ exactly, but instead an upper bound A, such that $\gamma _{\mathrm {Frob}} \leqslant A \leqslant M \gamma _{\mathrm {Frob}}$ , for some fixed $M \geqslant 1$ , the algorithm works as well, but the bound on the average number of continuation steps is multiplied by M (see Remark I.28):

(2.18) $$ \begin{align} {\mathbb{E}}[K] \leqslant 9,000 n^3 M \, \Gamma(F). \end{align} $$

To obtain an interesting complexity result for a given class of unitarily invariant distributions of polynomial systems F, based on numerical continuation along rigid paths and Inequality (2.18), we need, firstly, to specify how to compute or approximate $\gamma _{\mathrm {Frob}}$ at a reasonable cost, and secondly, to estimate the expectation of $\Gamma (F)$ over F. For the application to dense Gaussian systems, considered in Part I, $\gamma _{\mathrm {Frob}}$ is computed directly, using the monomial representation of the system to compute all higher derivatives, and the estimation of $\Gamma (F)$ is mostly standard. Using the monomial representation is not efficient anymore in the black-box model. We will rely instead on a probabilistic estimation of $\gamma _{\mathrm {Frob}}$ , within a factor $\operatorname {poly}(n, {D})$ . However, this estimation may fail with small probability, compromising the correctness of the result.

3.1 Weyl norm

We recall, here, how to characterise the Weyl norm of a homogeneous polynomial as an expectation, which is a key observation behind Algorithm GammaProb to approximate $\gamma _{\mathrm {Frob}}(f, z)$ by random sampling.

Let $f \in \mathbb {C}[z_0,\dotsc ,z_n]$ be a homogeneous polynomial of degree ${D}> 0$ . In the monomial basis, f decomposes as $\sum _\alpha c_\alpha z^\alpha $ , where $\alpha = (\alpha _0,\dotsc ,\alpha _n)$ is a multi-index. The Weyl norm of f is defined as

(3.1) $$ \begin{align} \|f\|_W^2\, {\stackrel{.}{=}} \,\sum_\alpha \frac{\alpha_0! \dotsb \alpha_n!}{{D}!} {\left| {c_\alpha} \right|}^2. \end{align} $$

The following statement seems to be classical.

Proof. Let H be the space of homogeneous polynomials of degree ${D}$ in $z_0,\dotsc ,z_n$ . Both left-hand and right-hand sides of the first stated equality define a norm on H coming from a Hermitian inner product. The monomial basis is orthogonal for both: this is obvious for Weyl’s norm. For the $L^2$ -norm, this is [Reference Rudin37, Proposition 1.4.8]. So it only remains to check that the claim holds true when f is a monomial. By [Reference Rudin37, Proposition 1.4.9(2)], if $w^{\alpha } = w_0^{\alpha _0}\cdots w_n^{\alpha _n}$ is a monomial of degree ${D}$ , we have

(3.2) $$ \begin{align} \mathbb{E}\left[{\left| {w^\alpha} \right|}^2 \right] &= \frac{(n+1)! \alpha_0!\dotsb \alpha_n!}{(n+1+{D}){!}} = \frac{(n+1)! {D}!}{(n+1+{D})!} \cdot \frac{\alpha_0! \dotsb \alpha_n!}{{D}!}, \end{align} $$

(3.3) $$ \begin{align} &= \tbinom{n+1+{D}}{{D}}^{-1} \left\| w^\alpha \right\|^2_W, \end{align} $$

which is the claim. The second equality follows similarly from [Reference Rudin37, Proposition 1.4.9(1)].

The following inequalities will also be useful.

Lemma 3.2. For any homogeneous polynomial $f \in \mathbb {C}[z_0,\dotsc ,z_n]$ of degree ${D}$ ,

$$ \begin{align*} \binom{n+{D}}{{D}}^{-1} \|f\|^2_W \leqslant \max_{z\in {\mathbb{S}}(\mathbb{C}^{n+1})} {\left| {f(z)} \right|}^2 = \max_{w\in B(\mathbb{C}^{n+1})} {\left| {f(w)} \right|}^2 \leqslant \|f\|_W^2. \end{align*} $$

Proof. The first inequality follows directly from the second equality of Lemma 3.1. It is clear that the maximum is reached on the boundary. For the second inequality, we may assume (because of the unitary invariance of $\|-\|_W$ ) that the maximum of $|f|$ on the unit ball is reached at $(1,0,\dotsc ,0)$ . Besides, the coefficient $c_{{D},0,\dotsc ,0}$ of f is $f(1,0,\dotsc ,0)$ . Therefore,

$$\begin{align*}\max_{w\in B} |f(w)|^2 = {\left| {f(1,0,\dotsc,0)} \right|}^2 = {\left| {c_{{D},0,\dotsc,0}} \right|}^2 \leqslant \|f\|_W^2. \\[-40pt]\end{align*}$$

3.2 Probabilistic evaluation of the gamma number

The main reason for introducing the Frobenius norm in the $\gamma $ number, instead of the usual operator norm, is the equality (Lemma I.30)

(3.4) $$ \begin{align} \frac{1}{k!} \left\| \mathrm{d} _z^kf \right\|_{\mathrm{Frob}} = \|f(z+\bullet)_k\|_W,\\[-15pt]\nonumber \end{align} $$

where $\|f(z+\bullet )_k\|_W$ is the Weyl norm of the homogeneous component of degree k of the shifted polynomial $x\mapsto f(z+x)$ . It follows that

(3.5) $$ \begin{align} \gamma_{\mathrm{Frob}}(f, z) = \sup_{k\geqslant 2} \left( \left\| \mathrm{d} _zf \right\|^{-1} \|f(z+\bullet)_k\|_W \right)^{\frac{1}{k-1}}.\\[-15pt]\nonumber \end{align} $$

This equality opens up interesting ways for estimating $\gamma _{\mathrm {Frob}}$ , and, therefore, $\gamma $ . We used it to compute $\gamma _{\mathrm {Frob}}$ efficiently when f is a dense polynomial given in the monomial basis (see Section I.4.3.3). In that context, we would compute the shift $f(z+\bullet )$ in the same monomial basis in quasilinear time as $\min (n,{D})\to \infty $ . From there, the quantities $ \|f(z+\bullet )_k\|_W$ can be computed in linear time. In the black-box model, however, the monomial expansions (of either f or $f(z+\bullet )$ ) cannot fit into a $\operatorname {poly}(n, {D}) L(f)$ complexity bound, because the number of monomials of degree ${D}$ in $n+1$ variables is not $\operatorname {poly}(n,{D})$ . Nonetheless, we can obtain a good enough approximation of $\|f(z+\bullet )_k\|_W$ with a few evaluations but a nonzero probability of failure. This is the purpose of Algorithm 2, which we analyse in the next theorem.

Theorem 3.3. Given $f \in \mathbb {C}[x_0,\dotsc ,x_n]$ as a black-box function, an upper bound ${D}$ on its degree, a point $z \in \mathbb {C}^{n+1}$ and some $\varepsilon> 0$ , Algorithm 2 (GammaProb) computes some $\Gamma \geqslant 0$ , such that

$$\begin{align*}\gamma_{\mathrm{Frob}}(f, z) \leqslant \Gamma \leqslant 192n^2{D} \cdot \gamma_{\mathrm{Frob}}(f, z)\\[-15pt] \end{align*}$$

with probability at least $1-\varepsilon $ , using $O\left ( {D} \log \left (\frac {D}\varepsilon \right ) (L(f) + n + \log {D})\right )$ operations.

Moreover, for any $t \ge 1$ ,

$$\begin{align*}\mathbb{P} \left[ \Gamma \leqslant \frac{\gamma_{\mathrm{Frob}}(f, z)}{t} \right] \leqslant \varepsilon ^{1+ \frac12 \log_2 t}.\\[-15pt] \end{align*}$$

Note that we currently do not know how to estimate $\gamma _{\mathrm {Frob}}$ within an arbitrarily small factor. The key in Theorem 3.3 is to write each $\|f(z+\bullet )_k\|_W^2$ as an expectation (this is classical, see Section 3.1) and to approximate it by sampling (there are some obstacles). We assume that $z = 0$ by changing f to $f(z + \bullet )$ , which is harmless because the evaluation complexity is changed to $L(f) + O(n)$ . Furthermore, the homogeneous components $f_k$ of f are accessible as black-box functions; this is the content of the next lemma.

Proof. We first compute all $f(\xi ^i w)$ , for $0\leqslant i \leqslant {D}$ for some primitive root of unity $\xi $ of order ${D}+1$ . This takes $({D}+1) L(f) + O({D} n)$ arithmetic operations. Since

(3.6) $$ \begin{align} f(\xi^i w) = \sum_{k=0}^{D} \xi^{i k} f_k(w), \end{align} $$

we recover the numbers $f_k(w)$ with the inverse Fourier transform,

(3.7) $$ \begin{align} f_k(w) = \frac{1}{{D}+1}\sum_{i = 0}^{D} \xi^{-ik} f(\xi^i w). \end{align} $$

We may assume that ${D}$ is a power of two ( ${D}$ is only required to be an upper bound on the degree of f), and the fast Fourier transform algorithm has an $O({D}\log {D})$ complexity bound to recover the $f_k(z)$ (with slightly more complicated formulas, we can also use $\xi = 2$ to keep close to the pure BSS model).

We now focus on the probabilistic estimation of $\|f_k\|_W$ via a few evaluations of $f_k$ . Let $B\, {\stackrel {.}{=}}\, {B({\mathbb {C}}^{n+1})}$ denote the Euclidean unit ball in $\mathbb {C}^{n+1}$ , and let $w \in B$ be a uniformly distributed random variable. By Lemma 3.1, we have

(3.8) $$ \begin{align} \|f_k\|_W^2 = \binom{n+1+k}{k} {\mathbb{E}} \left[ {\left| {f_k(w)} \right|}^2 \right]. \end{align} $$

The expectation in the right-hand side can be estimated with finitely many samples of ${\left | {f_k(w)} \right |}^2$ . To obtain a rigorous confidence interval, we study some statistical properties of ${\left | {f_k(w)} \right |}^2$ . Let $w_1,\dotsc ,w_s$ be independent uniformly distributed variables in B, and let

(3.9) $$ \begin{align} \hat{\textsf{m}}_k^2 \,{\stackrel{.}{=}}\, \frac1s \sum_{i=1}^s {\left| {f_k(w_i)} \right|}^2 \end{align} $$

denote their empirical mean. Let $\textsf {m}_k^2\, {\stackrel {.}{=}}\, {\mathbb {E}}[ \hat {\textsf {m}}^2_k ] = {\mathbb {E}}[ |f_k(w)|^2 ]$ be the mean that we want to estimate (note that both $\textsf {m}_k$ and $ \hat {\textsf {m}}_k$ depend on $f_k$ ; we suppressed this dependence in the notation).

The next proposition shows that $\hat {\textsf {m}}_k^2$ estimates $\textsf {m}_k^2$ within a $\operatorname {poly}(n,k)^{k}$ factor with very few samples. The upper bound is obtained by a standard concentration inequality (Hoeffding’s inequality). The lower bound is more difficult, and very specific to the current setting, because we need to bound $\textsf {m}_k^2$ away from zero with only a small number of samples. Concentration inequalities do not apply because the standard deviation may be larger than the expectation, so a confidence interval whose radius is comparable to the standard deviation (which is what we can hope for with a small number of samples) may contain negative values.

Proposition 3.5. For any $0\leqslant k \leqslant {D}$ , we have, with probability at least $1 - 2^{1-s}$ ,

where s is the number of samples.

Before proceeding with the proof, we state two lemmas, the principle of which comes from [Reference Ji, Kollar and Shiffman26, Lemma 8].

Lemma 3.6. Let $g \in {\mathbb {C}}[z]$ be a univariate polynomial of degree k, and let $c \in {\mathbb {C}}$ be its leading coefficient. For any $\eta> 0$ ,

$$\begin{align*}\operatorname{vol} \left\{ z\in \mathbb{C} \mathrel{}\middle|\mathrel{} {\left| {g(z)} \right|}^2 \leqslant \eta \right\} \leqslant \pi k \left({{\left| {c} \right|}^{-2}}{\eta}\right)^{\frac1k}. \end{align*}$$

Proof. Let $u_1,\dotsc ,u_k \in \mathbb {C}$ be the roots of g, with multiplicities, so that

(3.10) $$ \begin{align} g(z) = c (z-u_1)\dotsb(z-u_k). \end{align} $$

The distance of some $z\in \mathbb {C}$ to the set $S \,{\stackrel {.}{=}}\, \left \{ u_1,\dotsc ,u_k \right \}$ is the minimum of all ${\left | {z-u_i} \right |}$ . In particular

(3.11) $$ \begin{align} \operatorname{dist}(z, S)^k \leqslant \prod_{i=1}^k {\left| {z-u_i} \right|} = {\left| {c} \right|}^{-1} {\left| {g(z)} \right|}. \end{align} $$

Therefore,

(3.12) $$ \begin{align} \left\{ z\in \mathbb{C} \mathrel{}\middle|\mathrel{} {\left| {g(z)} \right|}^2 \leqslant \eta \right\} \subset \bigcup_{i=1}^k B \left( u_i, {\left| {c} \right|}^{-\frac1k} \eta^{\frac{1}{2k}} \right), \end{align} $$

where $B(u_i, r) \subseteq \mathbb {C}$ is the disk of radius r around $u_i$ . The volume of $B(u_i, r)$ is $\pi r^2$ , so the claim follows directly.

Lemma 3.7. If $w \in B$ is a uniformly distributed random variable, then for all $\eta> 0$ ,

$$\begin{align*}{\mathbb{P}} \left[ {\left| {f_k(w)} \right|}^2 \leqslant \eta \max_{\mathbb{S}} |f_k|^2 \right] \leqslant (n+1)k\eta^{\frac{1}{k}}, \end{align*}$$

where $\max _{\mathbb {S}} |f_k|$ is the maximum value of $|f_k|$ on the unit sphere in $\mathbb {C}^{n+1}$ .

Proof. Let c be the coefficient of $x_n^k$ in $f_k$ . It is the value of $f_k$ at $(0,\dotsc ,0,1)$ . Up to a unitary change of coordinates, $|f_k|$ reaches a maximum at $(0,\dotsc ,0,1)$ so that $c = \max _{\mathbb {S}} |f_k|$ . Up to scaling, we may further assume that $c=1$ . For any $(p_0,\ldots ,p_{n-1}) \in {\mathbb {C}}^{n}$ ,

(3.13) $$ \begin{align} \operatorname{vol} \left\{ z \in {\mathbb{C}} \mathrel{}\middle|\mathrel{} {\left| {f_k(p_0,\dotsc,p_{n-1}, z)} \right|}^2 \leqslant \eta \right\} \leqslant \pi k {\eta}^{1/k}, \end{align} $$

by Lemma 3.6 applied to the polynomial $g(z) = f_k(p_0,\dotsc ,p_{n-1}, z)$ , which, by construction, is monic. It follows, from the inclusion $B(\mathbb {C}^{n+1})\subseteq B(\mathbb {C}^n) \times \mathbb {C}$ , that

(3.14) $$ \begin{align} \operatorname{vol} \left\{ w\in B(\mathbb{C}^{n+1}) \mathrel{}\middle|\mathrel{} {\left| {f_k(w)} \right|}^2 \leqslant \eta \right\} \end{align} $$

(3.15) $$ \begin{align} &\leqslant \operatorname{vol}\left\{ (p_0,\dotsc,p_{n-1}, z)\in B(\mathbb{C}^{n}) \times \mathbb{C} \mathrel{}\middle|\mathrel{} {\left| {f_k(p_0,\dotsc,p_{n-1}, z)} \right|}^2 \leqslant \eta \right\} \end{align} $$

(3.16) $$ \begin{align} &\leqslant \operatorname{vol} B(\mathbb{C}^{n}) \cdot \pi k {\eta}^{\frac{1}{k}}. \end{align} $$

Using $\operatorname {vol} B(\mathbb {C}^{n}) = \frac {\pi ^n}{n!}$ and dividing both sides by $\operatorname {vol} B(\mathbb {C}^{n+1})$ concludes the proof.

Lemma 3.8. For any $\eta> 0$ , we have

Proof. Put $M \,{\stackrel {.}{=}}\, \max _{\mathbb {S}} {\left | {f_k} \right |}$ . If $\hat {\textsf {m}}_k^2 \le \eta M^2$ , then at least $\lceil s/2 \rceil $ samples among ${\left | {f(w_1)} \right |}^2,\dotsc ,{\left | {f(w_s)} \right |}^2$ satisfy ${\left | {f(w_i)} \right |}^2 \leqslant 2\eta M^2$ . By the union bound and Lemma 3.7, we obtain,

(3.17) $$ \begin{align} {\mathbb{P}} \left[ \hat{\textsf{m}}_k^2 \leqslant \eta M^2 \right] &\leqslant \binom{s}{\lceil s/2 \rceil} {\mathbb{P}} \left[ {\left| {f(w)} \right|}^2 \leqslant 2 \eta M^2 \right]^{\lceil s/2 \rceil} \end{align} $$

(3.18) $$ \begin{align} &\leqslant 2^s \left( (n+1)k \eta^{\frac1k} \right)^{\frac s2} \end{align} $$

(3.19) $$ \begin{align} &\leqslant \left( 8nk \eta^{\frac1k} \right)^{\frac{s}2}. \end{align} $$

To conclude, we note that $\textsf {m}_k \leqslant M$ .

Proof of Proposition 3.5

With $\eta \,{\stackrel {.}{=}}\, \left (32 nk \right )^{-k}$ , Lemma 3.8 gives

(3.20) $$ \begin{align} {\mathbb{P}} \left[ \hat{\textsf{m}}_k^2 \le \eta \textsf{m}_k^2 \right] \leqslant \left( {8} n k \eta^{\frac1k} \right)^{\frac s2} = 2^{-s}. \end{align} $$

It follows that

(3.21) $$ \begin{align} \mathbb{P} \left[ \textsf{m}_k^2 \leqslant \left({32} n k\right)^{k} \hat{\textsf{m}}_k^2 \right] \geqslant 1-2^{-s}, \end{align} $$

which is the stated left-hand inequality.

For the right-hand inequality, we apply Hoeffding’s inequality, (e.g., [Reference Boucheron, Lugosi and Massart10, Theorem 2.8]). The variable $s \hat {\textsf {m}}_k^2$ is a sum of s independent variables lying in the interval $[0, M^2]$ , where we, again, abbreviate $M \,{\stackrel {.}{=}}\, \max _{\mathbb {S}} {\left | {f_k} \right |}$ . Accordingly, for any $C \geqslant 1$ ,

(3.22) $$ \begin{align} {\mathbb{P}} \left[ \hat{\textsf{m}}_k^2 \geqslant C \textsf{m}_k^2 \right] = {\mathbb{P}} \left[ s \hat{\textsf{m}}_k^2 - s \textsf{m}^2_k \geqslant (C-1) s \textsf{m}^2_k \right] \leqslant \exp \left( - \frac{2 (C-1)^2 s^2 \textsf{m}_k^4}{sM^4} \right). \end{align} $$

By Lemma 3.2 combined with (3.8), we have

(3.23) $$ \begin{align} M^2 \le \binom{n+1+k}{k} \textsf{m}^2_k. \end{align} $$

Applying this bound, we obtain

(3.24) $$ \begin{align} {\mathbb{P}} \left[ \hat{\textsf{m}}_k^2 \geqslant C \textsf{m}_k^2 \right] \leqslant \exp \left( - \frac{2 (C-1)^2 s}{\binom{n+1+k}{k}^2} \right). \end{align} $$

We choose $C = (6n)^k$ and simplify further using the inequality $\binom {m+k}{k} \le \frac {(m+k)^k}{k!} \le (e(m+k)/k)^k$ and $e(n+1+k)/k \le e(n+3)/2$ (use $k\ge 2$ ) to obtain

(3.25) $$ \begin{align} \frac{C-1}{\binom{n+1+k}{k}} &\geqslant \frac{(6n)^k - 1 }{\left( \frac{e (n+3)}{2} \right)^{k}} \geqslant \left( \frac{12 n}{e (n+3)} \right)^{k} - \left( \frac{2}{e(n+3)} \right)^{k} \end{align} $$

(3.26) $$ \begin{align} &\geqslant \left( \frac{3}{e} \right)^2 - \left( \frac{2}{4 e} \right)^2 \geqslant \sqrt{\frac12 \log{2}}. \end{align} $$

We obtain, therefore

(3.27) $$ \begin{align} {\mathbb{P}} \left[ \hat{\textsf{m}}_k^2 \geqslant (6n)^k \textsf{m}_k^2 \right] \leqslant \exp (-\log(2) s) = 2^{-s}. \end{align} $$

Combined with (3.21), the union bound implies

(3.28) $$ \begin{align} {\mathbb{P}} \left[ \hat{\textsf{m}}_k^2 \leqslant (32nk)^{-k} \textsf{m}_k^2 \mbox{ or } \hat{\textsf{m}}_k^2 \ge (6n)^k \textsf{m}_k^2 \right] \leqslant 2 \cdot 2^{-s} \end{align} $$

and the proposition follows.

Proof of Theorem 3.3

Recall that we assume that $z=0$ . Proposition 3.5 can be rephrased as follows: with probability at least $1-2^{1-s}$ , we have

(3.29) $$ \begin{align} \textsf{m}_k^2 \le (32nk)^{k} \hat{\textsf{m}}_k^2 \leqslant (192n^2 k)^k \textsf{m}_k^2. \end{align} $$

Defining

(3.30) $$ \begin{align} c^2_k\, {\stackrel{.}{=}}\, \binom{n+1+k}{k} (32 n k)^k \hat{\textsf{m}}^2_k , \end{align} $$

using that by (3.8)

(3.31) $$ \begin{align} \|f_k\|_W^2 = \binom{n+1+k}{k} \textsf{m}_k^2 , \end{align} $$

and applying the union bound, we, therefore, see that

(3.32) $$ \begin{align} \|f_k\|^2_W \leqslant c^2_k \leqslant {(192 n^2 k)^k} \cdot \|f_k\|^2_W \end{align} $$

holds for all $2\le k\le {D}$ , with probability at least $1-{D} 2^{1-s}$ . If we chose $s = \left \lceil 1 + \log _2 \frac {D}\varepsilon \right \rceil $ , then ${D} 2^{1-s} \leqslant \varepsilon $ . Recall from (3.4) and (3.5) that

(3.33) $$ \begin{align} \gamma_{\mathrm{Frob}}(f, z) = \max_{{D} \ge k\geqslant 2} \left( \left\| \mathrm{d} _0 f \right\|^{-1} \|f_k \|_W \right)^{\frac{1}{k-1}}. \end{align} $$

Noting that $(192 n^2 k)^{\frac {k}{k-1}} \leqslant (192 n^2 {D})^2$ , for $2\leqslant k\leqslant {D}$ , we conclude that the random variable

(3.34) $$ \begin{align} \Gamma\, {\stackrel{.}{=}} \,\max_{2 \leqslant k \leqslant {D}} \left( \|\mathrm{d} _0 f\|^{-1} c_k \right)^{\frac{1}{k-1}}, \end{align} $$

which is returned by Algorithm 2, indeed satisfies

(3.35) $$ \begin{align} \gamma_{\mathrm{Frob}} (f,z) \le \Gamma \le 192 n^2 {D} \cdot \gamma_{\mathrm{Frob}} (f,z) \end{align} $$

with probability at least $1-\varepsilon $ , which proves the first assertion.

For the assertion on the number of operations, it suffices to note that by Lemma 3.4, the computation of the derivative $d_0f$ and of $\hat {\textsf {m}}_2,\dotsc ,\hat {\textsf {m}}_{D}$ can be done with $O(s{D} (L(f) + n + \log {D}))$ arithmetic operations.

It only remains to check, for any $t \geqslant 1$ , the tail bound

(3.36) $$ \begin{align} \mathbb{P} \left[ \Gamma \leqslant \frac{\gamma_{\mathrm{Frob}} (f, z)}{t} \right] \leqslant \varepsilon ^{1+\frac12 \log_2 t}. \end{align} $$

Unfolding the definitions (3.33) and (3.34) and using, again, (3.31), we obtain

(3.37) $$ \begin{align} \mathbb{P} \left[ \Gamma \leqslant \frac{\gamma_{\mathrm{Frob}} (f, z)}{t} \right] &\leqslant \sum_{k=2}^{D} \mathbb{P} \left[ (32nk)^k \hat{\textsf{m}}_k^2 \leqslant t^{-2 (k-1)} \textsf{m}_k^2 \right] \end{align} $$

(3.38) $$ \begin{align} &\leqslant \sum_{k=2}^{D} \left( 8 nk \cdot \left( (32nk)^k t^{2(k-1)} \right)^{-\frac1k} \right)^{\frac s2}, \qquad \text{by Lemma~3.8}, \end{align} $$

(3.39) $$ \begin{align} &= \sum_{k=2}^{D} \left( \frac14 t^{-2\frac{k-1}{k}} \right)^{\frac s2} \leqslant {D} 2^{-s} t^{-\frac s2}. \end{align} $$

Since $s = \left \lceil 1 + \log _2 \frac {D}\varepsilon \right \rceil $ , we have ${D} 2^{-s} \leqslant \varepsilon $ . Furthermore, $s \geqslant -\log _2 \varepsilon $ , so

(3.40) $$ \begin{align} t^{-\frac s2} \leqslant t^{\frac 12 \log_2 \varepsilon } = \varepsilon ^{\frac12 \log_2 t}, \end{align} $$

which proves (3.36).

3.3 A Monte-Carlo continuation algorithm

We deal, here, with the specifics of a numerical continuation with a step-length computation that may be wrong.

The randomised algorithm for the evaluation of the step length can be plugged into the rigid continuation algorithm (Algorithm 1). There is no guarantee, however, that the randomised computations of the $\gamma _{\mathrm {Frob}}$ fall within the confidence interval described in Theorem 3.3 and, consequently, there is no guarantee that the corresponding step-length estimation is accurate. If step lengths are underestimated, we don’t control any more the complexity: as the step lengths go to zero, the number of steps goes to infinity. Overestimating a single step length, instead, may undermine the correctness of the result, and the subsequent behaviour of the algorithm is unknown (it may even not to terminate). So we introduce a limit on the number of continuation steps. Algorithm 3 is a corresponding modification of Algorithm 1. When reaching the limit on the number of steps, this algorithm halts with a failure notification.

In case Algorithm 3 fails, it is natural to restart the computation with a higher iteration limit. This is Algorithm 4. We can compare its complexity to that of Algorithm 1, which assumes an exact computation of $\gamma $ . Let $K(F, {\mathbf u}, {\mathbf v}, \zeta )$ be a bound for the number of iterations performed by Algorithm 1 on input F, ${\mathbf u}$ , ${\mathbf v}$ and $\zeta $ , allowing an overestimation of the step length up to a factor $192 n^2{D}$ (in view of Theorem 3.3).

Proof. Let $K\,{\stackrel {.}{=}}\, K(F, {\mathbf u}, {\mathbf v}, \zeta )$ . By definition of K, if all approximations lie in the desired confidence interval, then BoundedBlackBoxNC terminates after at most K iterations. So as soon as ${K_{\mathrm {max}}} \geqslant K$ , BoundedBlackBoxNC may return Fail only if the approximation of some $\gamma _{\mathrm {Frob}}$ is not correct. This happens with probability at most $\varepsilon $ at each iteration of the main loop in Algorithm 4, independently. So the number of iterations is finite, almost surely. That the result is correct with probability at least $1-\varepsilon $ follows from Proposition 3.9.

We now consider the total cost. At the mth iteration, we have ${K_{\mathrm {max}}}=2^m$ , so the cost of the mth iteration is $\operatorname {poly}(n, {D}) \cdot 2^m \log (2^m \varepsilon ^{-1}) \cdot L(f)$ , by Proposition 3.9. Put $\ell \,{\stackrel {.}{=}}\, \lceil \log _2 K \rceil $ . If the mth iteration is reached for some $m> \ell $ , then all the iterations from $\ell $ to $m-1$ have failed. This has a probability $\leqslant \varepsilon ^{m - \ell }$ to happen, so, if I denotes the number of iterations, we have

(3.41) $$ \begin{align} \mathbb{P}[I \geqslant m] \leqslant \min\left(1, \varepsilon ^{m - \ell}\right).\\[-15pt]\nonumber \end{align} $$

The total expected cost is, therefore, bounded by

(3.42) $$ \begin{align} \mathbb{E}[\mathrm{cost}] &\leqslant \operatorname{poly}(n, {D}) L(f)\sum_{m=1}^\infty 2^m \log(2^m \varepsilon ^{-1}) \mathbb{P}[I \geqslant m] \end{align} $$

(3.43) $$ \begin{align} &\leqslant \operatorname{poly}(n, {D}) L(f)\sum_{m=1}^\infty 2^m \log (2^m \varepsilon ^{-1}) \min\left(1, \varepsilon ^{m-\ell}\right). \\[-15pt]\nonumber\end{align} $$

The claim follows easily from splitting the sum into two parts, $1\leqslant m < \ell $ and $m> \ell $ , and applying the bounds (with $c=\log \varepsilon ^{-1}$ )

(3.44) $$ \begin{align} \sum_{m=1}^{\ell-1} 2^m (m + c) \leqslant (\ell+c)2^\ell \\[-15pt]\nonumber\end{align} $$

and, for $\varepsilon \in (0,\frac 14)$ ,

(3.45) $$ \begin{align} \sum_{m=\ell}^\infty 2^m (m+c) \varepsilon ^{m-\ell} \leqslant \frac{(\ell+c) 2^\ell}{(1-2\varepsilon )^2} \leqslant 4 (\ell+c) 2^\ell. \end{align} $$

4.1 Complexity of sampling the rigid solution variety

Towards the proof of Theorems 1.2 and 4.2, we first review the complexity of sampling the initial pair for the numerical continuation. In the rigid setting, this sampling boils down to sampling hypersurfaces, which, in turn, amounts to computing roots of univariate polynomials (see Part I, Section 2.4). Some technicalities are required to connect known results about root-finding algorithms to our setting, and especially the parameter $\Gamma (F)$ , but the material is very classical. Below, $\log \log (x)$ is to be interpreted as the function $\log _2\log _2(x+2)$ which is defined for all positive x. Since our statements are asymptotic, this does not change anything.

Proof. Following Corollary I.9, we can compute a uniformly distributed zero of f by first sampling a line $\ell \subset \mathbb {P}^n$ uniformly distributed in the Grasmannian of lines, and then sampling a uniformly distributed point in the finite set $\ell \cap V(f)$ . To do this, we consider the restriction $f|_\ell $ , which, after choosing an orthonormal basis of $\ell $ , is a bivariate homogeneous polynomial, and compute its roots. The representation of $f|_\ell $ in a monomial basis can be computed by ${D}+1$ evaluations of f and interpolation, at a cost $O \left ( {D} (L(f) + n + \log {D}) \right )$ , as in Lemma 3.4. By Lemma 4.5 below, computing the roots takes

(4.1) $$ \begin{align} \operatorname{poly}({D}) \log\log \left(\max_{\zeta \in \ell\cap V(f)} \gamma(f|_\ell, \zeta)\right) \end{align} $$

operations on average. We assume a 6th type of node to refine approximate roots into exact roots (recall the discussion in Section 1.3). Then we have, by the definition (1.5) of $\Gamma (f|_\ell )$ ,

(4.2)

By Jensen’s inequality [Reference Bürgisser and Cucker15, Proposition 2.28], using the concavity of $\log _2\log _2(x)$ on $\lbrack 2,\infty )$ , we obtain

(4.3) $$ \begin{align} {\mathbb{E}}_{\ell} \left[ \log_2\log_2\left( 2 + {D}\, \Gamma(f|_\ell)^2 \right) \right] \leqslant \log_2\log_2 \left(2 + {D} {\mathbb{E}}_{\ell} \left[\Gamma(f|_\ell)^2 \right]\right). \end{align} $$

Finally, Lemma 4.6 below gives

(4.4) $$ \begin{align} \log_2\log_2 \left(2 + {D} {\mathbb{E}}_{\ell} \left[\Gamma(f|_\ell)^2 \right]\right) \leqslant \log_2 \log_2 \left( 2+ 2n{D} \, \Gamma(f)^2 \right) \end{align} $$

and the claim follows.

Proof. The proof essentially relies on the following known fact due to [Reference Renegar35] (see also [Reference Pan33, Thm. 2.1.1 and Cor. 2.1.2], for tighter bounds). Let $f \in \mathbb {C}[t]$ be a given polynomial of degree ${D}$ , $R> 0$ be a known upper bound on the modulus of the roots $\xi _1,\dotsc ,\xi _{D} \in \mathbb {C}$ of f and $\varepsilon>0$ . We can compute from these data with $\operatorname {poly}({D}) \log \log \frac R\varepsilon $ operations approximations $x_1,\dotsc ,x_n \in \mathbb {C}$ of the zeros, such that ${\left | {\xi _i - x_i} \right |} \leqslant \varepsilon $ .

To apply this result to the given homogeneous polynomial g, we first apply a uniformly random unitary transformation $u\in U(2)$ to the given g and dehomogenise $u\cdot g$ , obtaining the univariate polynomial $f\in {\mathbb {C}}[t]$ .

We first claim that, with probability at least $3/4$ , we have: ( $*$ ) $|\xi _i| \le 2 \sqrt {{D}}$ for all zeros $\xi _i\in {\mathbb {C}}$ of f. This can be seen as follows. We measure distances in ${\mathbb {P}}^1$ with respect to the projective (angular) distance. The disk of radius $\theta $ around a point in ${\mathbb {P}}^1$ , has measure at most $\pi (\sin \theta )^2$ [Reference Bürgisser and Cucker15, Lemma 20.8]. Let $\sin \theta =(2\sqrt {{D}})^{-1}$ . Then a uniformly random point p in ${\mathbb {P}}^1$ lies in a disk of radius $\theta $ around a root of f with probability at most ${D} (\sin \theta )^2 \le 1/4$ . Write $0\,{\stackrel {.}{=}} \,[1:0]$ and $\infty \,{\stackrel {.}{=}}], [0:1]$ , and note that $\operatorname {dist}(0,p) + \operatorname {dist}(p, \infty ) =\pi /2$ for any $p\in {\mathbb {P}}^1$ . Since $u^{-1}(\infty )$ is uniformly distributed, we conclude that with probability at least $3/4$ , each zero $\zeta \in {\mathbb {P}}^1$ of g satisfies $\operatorname {dist}(\zeta , u^{-1}(\infty )) \ge \theta $ , which means $\operatorname {dist}(\zeta , u^{-1}( 0)) \le \pi /2 -\theta $ . The latter easily implies for the corresponding affine root $\xi =\zeta _1/\zeta _0$ of f that $|\xi | \le (\tan \theta )^{-1} \le (\sin \theta )^{-1}= 2\sqrt {{D}}$ , hence, ( $*$ ) holds.

The maximum norm of a zero of $f\in {\mathbb {C}}[{t}]$ can be computed with a small relative error with $O({D} \log {D})$ operations [Reference Pan34, Fact 2.2(b)], so we can test the property ( $*$ ). We repeatedly sample a new $u \in U(2)$ until ( $*$ ) holds. Each iteration succeeds with probability at least $\frac 34$ of success, so there are at most two iterations on average.

For a chosen $\varepsilon>0$ , we can now compute with Renegar’s algorithm the roots of f, up to precision $\varepsilon $ with $\operatorname {poly}({D}) \log \log \frac {1}{\varepsilon }$ operations (where the $\log \log 2\sqrt {{D}}$ is absorbed by $\operatorname {poly}({D})$ ). By homogenising and transforming back with $u^{-1}$ , we obtain approximations $p_1,\dotsc ,p_{D}$ of the projective roots $\zeta _1,\ldots ,\zeta _{D}$ of g up to precision $\varepsilon $ , measured in projective distance.

The remaining difficulty is that the $p_i$ might not be approximate roots of g, in the sense of Smale. However, suppose that for all i we have

(4.5) $$ \begin{align} \varepsilon \gamma(g,p_i) \leqslant \tfrac{1}{11}. \end{align} $$

Using that $z \mapsto \gamma (g,z)^{-1}$ is 5-Lipschitz continuous on $\mathbb {P}^1$ (by Lemma I.31), we see that $\varepsilon \gamma (g, \zeta _i) \leqslant \frac 16$ for all i. This is known to imply that $p_i$ is an approximate zero of $p_i$ ([Reference Shub and Smale40], and Theorem I.12 for the constant). On the other hand, using again the Lipschitz property, we are sure that Condition (4.5) is met as soon as $\varepsilon \gamma _{\max } \leqslant \tfrac {1}{16}$ .

So starting with $\varepsilon = \frac 12$ , we compute points $p_1,\dotsc ,p_{D}$ approximating $\zeta _1,\dotsc ,\zeta _{D}$ up to precision $\varepsilon $ until (4.5) is met for all $p_i$ , squaring $\varepsilon $ after each unsuccessful iteration. Note that Renegar’s algorithm need not be restarted when $\varepsilon $ is refined. We have $\varepsilon \gamma _{\max } \leqslant \tfrac {1}{16}$ after at most $\log \log ( 16\gamma _{\max } )$ iterations. Finally, note that we do not need to compute exactly $\gamma $ , an approximation within factor 2 is enough, with appropriate modifications of the constants, and this is achieved by $\gamma _{\mathrm {Frob}}$ , see (1.4), which we can compute in $\operatorname {poly}({D})$ operations.

Proof. Let $\ell \subseteq \mathbb {P}^n$ be a uniformly distributed random projective line, and let $\zeta \in \ell $ be uniformly distributed among the zeros of $f|_\ell $ . Then $\zeta $ is also a uniformly distributed zero of f (see Corollary I.9). Let $\theta $ denote the angle between the tangent line $T_{\zeta } \ell $ and the line $T_{\zeta } V(f)^{\perp }$ normal to $V(f)$ at $\zeta $ . By an elementary geometric reasoning, we have $\|\mathrm {d} _{\zeta } f|_\ell \| = \|\mathrm {d} _{\zeta } f\| \cos \theta $ . Moreover, $\| \mathrm {d} _{\zeta }^k f|_\ell \|_{\mathrm {Frob}} \leqslant \| \mathrm {d} ^k_{\zeta } f\|_{\mathrm {Frob}}$ . So it follows that

(4.6) $$ \begin{align} \gamma_{\mathrm{Frob}}(f|_\ell,\zeta)^2 \leqslant \gamma_{\mathrm{Frob}}(f, \zeta)^2\cos(\theta)^{-2}. \end{align} $$

In order to bound this, we consider now a related but different distribution. As above, let $\zeta $ be a uniformly distributed zero of f. Consider now a uniformly distributed random projective line $\ell '$ passing through $\zeta $ . The two distributions $(\ell ,\zeta )$ and $(\zeta ,\ell ')$ are related by Lemma I.5 as follows: for any integrable function h of $\ell $ and $\zeta $ , we have

(4.7) $$ \begin{align} \mathbb{E}_{\ell,\zeta} [ h(\ell, \zeta) ] = c\, \mathbb{E}_{\zeta,\ell'} [ h(\ell', \zeta) {\operatorname{det}\nolimits}^\perp ({T}_{\zeta} \ell', {T}_{\zeta} V(f)) ], \end{align} $$

where c is some normalisation constant and where ${\operatorname {det}\nolimits }^\perp ({T}_{\zeta } \ell ', {T}_{\zeta } V(f))$ is defined in I, Section 2.1. It is only a matter of unfolding definitions to see that it is equal to $\cos \theta '$ , where $\theta '$ denotes the angle between $T_{\zeta } \ell '$ and $T_{\zeta } V(f)^\perp $ . With $h = 1$ , we obtain $c = \mathbb {E} \left [ \cos \theta ' \right ]^{-1}$ and, therefore, we get

(4.8) $$ \begin{align} \mathbb{E}_{\ell, \zeta} [ h(\ell, \zeta) ] = \mathbb{E}_{\zeta, \ell'} [ h(\ell', \zeta) \cos\theta' ] \, \mathbb{E}[\cos \theta']^{-1}. \end{align} $$

We analyse now the distribution of $\theta '$ : $\cos (\theta ')^2$ is a beta-distributed variable with parameters $1$ and $n-1$ : indeed, $\cos (\theta ')^2 = {\left | {u_1} \right |}^2 / \|u\|^2$ where $u\in \mathbb {C}^n$ is a Gaussian random vector, and it is well known that the distribution of this quotient of $\chi ^2$ -distributed random variables is a beta-distributed variable. Generally, the moments of a beta-distributed random variable Z with parameters $\alpha ,\beta $ satisfy

(4.9) $$ \begin{align} \mathbb{E}[ Z^{r} ] = \frac{B(\alpha + r, \beta)}{B(\alpha,\beta)} , \end{align} $$

where B is the Beta function and $r>-\alpha $ . In particular, for $r> -1$ ,

(4.10) $$ \begin{align} \mathbb{E}_{\zeta, \ell'}[ \cos(\theta')^{2r} ] = \frac{B(1+r, n-1)}{B(1,n-1)} , \end{align} $$

and, hence

(4.11) $$ \begin{align} \mathbb{E} \left[ \cos(\theta')^{-1} \right] \mathbb{E} \left[ \cos(\theta') \right]^{-1} = \frac{B(\frac12, n-1)}{B(\frac32, n-1)} = 2n - 1. \end{align} $$

Continuing with (4.8), we obtain

(4.12) $$ \begin{align} \hspace{12pt}\mathbb{E}_\ell \left[ \Gamma(f|_\ell)^2 \right] &= \mathbb{E}_{\ell, \zeta} \left[ \gamma_{\mathrm{Frob}}(f|_\ell, \zeta)^2 \right],\ \,\qquad\qquad\qquad\quad\qquad\qquad \text{by~(1.5),}\qquad\qquad \end{align} $$

(4.13) $$ \begin{align} &\leqslant \mathbb{E}_{\ell, \zeta} \left[ \gamma_{\mathrm{Frob}}(f, \zeta)^2 \cos(\theta)^{-2} \right],\,\qquad\qquad\qquad\qquad \text{by~(4.6),} \end{align} $$

(4.14) $$ \begin{align} &= \mathbb{E}_{\zeta, \ell'} \left[ \gamma_{\mathrm{Frob}}(f, \zeta)^2 \cos(\theta')^{-1} \right] \mathbb{E} \left[ \cos(\theta') \right]^{-1}, \qquad\text{by~(4.8),} \end{align} $$

(4.15) $$ \begin{align} &= \mathbb{E}_{\zeta} \left[ \gamma_{\mathrm{Frob}}(f, \zeta)^2 \right] \mathbb{E}_{\ell'} \left[\cos(\theta')^{-1} \right] \mathbb{E} \left[ \cos(\theta') \right]^{-1} \end{align} $$

(4.16) $$ \begin{align} &= \Gamma(f)^2 (2n-1), \qquad\qquad\qquad\qquad\qquad\qquad\qquad \text{by (4.11)} \end{align} $$

the second last equality (4.15) since the random variable $\theta '$ is independent from $\zeta $ . This concludes the proof.

4.2 Proof of Theorem 4.2

We now study the average complexity of the Algorithm , where ${\mathbf u} \in {\mathcal {U}}$ is uniformly distributed. Recall that $\Gamma ({\mathbf u}\cdot F)= \Gamma (F)$ , by unitary invariance of $\gamma _{\mathrm {Frob}}$ , and $L({\mathbf u} \cdot F) = L(F) + O(n^3)$ .

The sampling operation costs at most $\operatorname {poly}(n, {D}) \cdot L(F) \cdot \log \log \Gamma (F)$ on average, by Proposition 4.3. The expected cost of the continuation phase is $\operatorname {poly}(n, {D}) \cdot L(F) \cdot K (\log K +\log \varepsilon ^{-1})$ , by Proposition 3.10, where $K=K({\mathbf u} \cdot F, \mathbf {1}_{{\mathcal {U}}}, {\mathbf v}, \zeta )$ and $({\mathbf v}, \zeta )$ is the sampled initial pair. By unitary invariance,

(4.17) $$ \begin{align} K({\mathbf u}\cdot F, \mathbf{1}_{{\mathcal{U}}}, {\mathbf v}, \zeta) = K(F, {\mathbf u}, {\mathbf v}', \zeta), \end{align} $$

where ${\mathbf v}' = {\mathbf v}{\mathbf u}$ . Moreover, since ${\mathbf v}$ is uniformly distributed and independent from ${\mathbf u}$ , ${\mathbf v}'$ is also uniformly distributed and independent from ${\mathbf u}$ , and $\zeta $ is a uniformly distributed zero of ${\mathbf v}' \cdot F$ . So the following proposition concludes the proof of Theorem 4.2.

Sketch of proof

The first bound ${\mathbb {E}} \left [ K \right ] \leqslant \operatorname {poly}(n, {D}) \Gamma (F)$ was shown in Theorem I.27. Following mutatis mutandis the proof of Theorem I.27 (the only additional fact needed is Proposition 4.8 below for $a=3/2$ ), we obtain that

(4.18) $$ \begin{align} {\mathbb{E}} \left[ K^{\frac32} \right] \leqslant \operatorname{poly}(n, {D}) \Gamma(F)^{\frac32}. \end{align} $$

Next, we observe that the function $h : x \mapsto x^{\frac 23} (1+ \log x^{\frac 23})$ is concave on $[1,\infty )$ . By Jensen’s inequalities, it follows that

(4.19) $$ \begin{align} {\mathbb{E}} \left[ K \log K \right] \leqslant {\mathbb{E}} \left[ h( K^{\frac32} ) \right] \leqslant h \left( {\mathbb{E}} [K^{\frac32} ]\right) \leqslant \operatorname{poly}(n, {D}) \Gamma(F) \log \Gamma(F), \end{align} $$

which gives the claim.

The following statement extends Proposition I.17 to more general exponents. The proof technique is more elementary, and the result, although not as tight, good enough for our purpose.

Proposition 4.8. Let $M \in \mathbb {C}^{n\times (n+1)}$ be a random matrix whose rows are independent uniformly distributed vectors in $\mathbb {S}(\mathbb {C}^{n+1})$ , and let $\sigma _{\min }(M)$ be the smallest singular value of M. For all $a \in [1,2)$ ,

$$\begin{align*}{\mathbb{E}} \left[ \sigma_{\min}(M)^{-2a} \right] \leqslant \frac{n^{1+2a}}{2-a}, \end{align*}$$

and, equivalently with the notations of Proposition I.17,

$$\begin{align*}{\mathbb{E}} \left[ \kappa({\mathbf u}\cdot F, \zeta)^{2a} \right] \leqslant \frac{n^{1+2a}}{2-a}. \end{align*}$$

Proof. For short, let $\sigma $ denote $\sigma _{\min }(M)$ . Let $u_1,\dotsc ,u_n$ be the rows of M. By definition, there is a unit vector $x \in \mathbb {C}^n$ , such that

(4.20) $$ \begin{align} \| x_1 u_1 + \cdots + x_n u_n \|^2 = \sigma^2. \end{align} $$

If $V_i$ denotes the subspace of $\mathbb {C}^{n+1}$ spanned by all $u_j$ except $u_i$ , and $b_i$ denotes the squared Euclidean distance of $u_i$ to $V_i$ , then (4.20) implies $b_i \leqslant {\left | {x_i} \right |}^{-2} \sigma ^2$ for all i. Moreover, since x is a unit vector, there is at least one i, such that ${\left | {x_i} \right |}^2 \geqslant \frac {1}{n}$ . Hence, $n \sigma ^2 \geqslant \min _i b_i$ and, therefore

(4.21) $$ \begin{align} {\mathbb{E}} \left[ \sigma^{-2a} \right] \leqslant {n^a \mathbb{E} \left[\max_i b_i^{-a} \right]} \leqslant n^a \sum_{i=1}^n \mathbb{E}\left[b_i^{-a}\right]. \end{align} $$

To analyse the distribution of $b_i$ consider, for fixed $V_i$ , a standard Gaussian vector $p_i$ in $V_i$ , and an independent standard Gaussian vector $q_i$ in $V_i^\perp $ (note $\dim V_i^\perp =4$ ). Since $u_i$ is uniformly distributed in the sphere, it has the same distribution as $(p_i+q_i)/\sqrt {\|p_i\|^2 + \|q_i\|^2}$ . In particular, $b_i$ has the same distribution as $\|q_i\|^2/(\|p_i\|^2 + \|q_i\|^2)$ , which is a Beta distribution with parameters $2, n-1$ , since $\|p_i\|^2$ and $\|q_i\|^2$ are independent $\chi ^2$ -distributed random variables with $2n-2$ and $4$ degrees of freedom, respectively. By (4.9), we have for the moments, using $a<2$ ,

(4.22) $$ \begin{align} {\mathbb{E}} \left[ b_i^{-a} \right] = \frac{B(2-a,n-1) }{B(2,n-1)} = \frac{\Gamma(2-a) \Gamma(n+1)}{\Gamma(n+1-a)}. \end{align} $$

We obtain

(4.23) $$ \begin{align} {\mathbb{E}} \left[ b_i^{-a} \right] & = \frac{\Gamma(3-a)}{2-a} \cdot n \cdot \frac{\Gamma(n)}{\Gamma(n+1-a)}, && \text{using twice } \Gamma(x+1)=x\Gamma(x), \end{align} $$

(4.24) $$ \begin{align} &\leqslant \frac{1}{2-a} \cdot n \cdot n^{a-1}, && \text{by Gautschi's inequality.} \end{align} $$

In combination with (4.21), this gives the result.

4.3 Confidence boosting and proof of Theorem 1.2

We may leverage the quadratic convergence of Newton’s iteration to increase the confidence in the result of Algorithm 5 and reduce the dependence on $\varepsilon $ (the maximum probability of failure) from $\log \frac 1\varepsilon $ down to $\log \log \frac 1\varepsilon $ , so that we can choose $\varepsilon = \smash {10^{-10^{100}}}$ without afterthoughts, at least in the BSS model. On a physical computer, the working precision should be comparable with $\varepsilon $ , which imposes some limitations. A complete certification, without possibility of error, with $\operatorname {poly}(n,{D})$ evaluations of F, seems difficult to reach in the black-box model: with only $\operatorname {poly}(n,{D})$ evaluations, we cannot distinguish a polynomial system F from the infinitely many other systems with the same evaluations.

To describe this boosting procedure, we first recall some details about $\alpha $ -theory and Part I. Let $F\in {\mathcal {H}}$ be a polynomial system and $z\in \mathbb {P}^n$ be a projective point. Let ${\mathcal {N}}_F(z)$ denote the projective Newton iteration (and ${\mathcal {N}}^k_F(z)$ denote the composition of k projective Newton iterations). Let

(4.25) $$ \begin{align} \beta(F, z) \,{\stackrel{.}{=}}\, d_{\mathbb{P}} \left( z, {\mathcal{N}}_F(z) \right). \end{align} $$

There is an absolute constant $\alpha _0$ , such that for any $z\in \mathbb {P}^n$ , if $\beta (F, z) \gamma (F, z) \leqslant \alpha _0$ , then z is an approximate zero of F [Reference Dedieu and Shub17, Theorem 1] (one may take $\alpha _0= 1/137$ ). This is one of many variants of the $\alpha $ -theorem of [Reference Smale45]. There may be differences in the definition of $\gamma $ or $\beta $ , or even the precise definition of approximate zero, but they only change the constant $\alpha _0$ .

It is important to be slightly more precise about the output of Algorithm 5 (when all $\gamma $ ’s are correctly estimated, naturally, and hence, the output is an approximate zero of F): by the design of the numerical continuation (see Proposition I.22 with $C=15$ and $A=\frac {1}{4C}$ ), the output point $z^*\in \mathbb {P}^n$ satisfies

(4.26) $$ \begin{align} d_{\mathbb{P}}(z^*,\zeta) \hat{\gamma}_{\mathrm{Frob}}(F, \zeta) \leqslant \frac{1}{4\cdot 15} = \frac{1}{60}, \end{align} $$

for some zero $\zeta $ of F, where $\hat {\gamma }_{\mathrm {Frob}}$ is the split Frobenius $\gamma $ number (see Section 2.3). This implies (see Theorem I.12), using $\gamma \leqslant \hat {\gamma }_{\mathrm {Frob}}$ , that

(4.27) $$ \begin{align} d_{\mathbb{P}} \left( {\mathcal{N}}_F^k(z^*), \zeta \right) \leqslant 2^{1-2^k} d_{\mathbb{P}} (z^*, \zeta). \end{align} $$

The last important property we recall is the 15-Lipschitz continuity of the function $z\in \mathbb {P}^n \mapsto \hat {\gamma }_{\mathrm {Frob}}(F, z)^{-1}$ (Lemmas I.26 and I.31).

Algorithm 6 checks the criterion $\beta (F, z^*) \gamma (F, z^*) \leqslant \alpha _0$ after having refined to $z^*$ a presumed approximate zero z with a few Newton iterations. If the input point z is indeed an approximate zero, then $\beta (F, z^*)$ will be very small, and it will satisfy the criterion above even with a very gross approximation of $\gamma (F,z^*)$ .

Proof. We use the notations (k, $z^*$ and c) of Algorithm 6. Assume first that (4.26) holds for z and some zero $\zeta $ of F. By (4.27) and (4.26),

(4.28) $$ \begin{align} d_{\mathbb{P}}(z^*, \zeta) \hat{\gamma}_{\mathrm{Frob}}(F, \zeta) \leqslant \frac{1}{60}. \end{align} $$

Using the Lipschitz continuity and (4.28),

(4.29) $$ \begin{align} \hat{\gamma}_{\mathrm{Frob}}(F, z^*) \leqslant \frac{\hat{\gamma}_{\mathrm{Frob}}(F, \zeta)}{1-15 d_{\mathbb{P}}(z^*, \zeta) \hat{\gamma}_{\mathrm{Frob}}(F, \zeta)} \leqslant \frac43 \hat{\gamma}_{\mathrm{Frob}}(F, \zeta), \end{align} $$

and it follows from (4.27) and (4.26), again, that

(4.30) $$ \begin{align} \beta(F, z^*) \hat{\gamma}_{\mathrm{Frob}}(F, z^*) &\leqslant \frac43 \left(d_{\mathbb{P}}(z^*, \zeta) + d_{\mathbb{P}}({\mathcal{N}}_F(z^*), \zeta) \right) \hat{\gamma}_{\mathrm{Frob}}(F, \zeta) \end{align} $$

(4.31) $$ \begin{align} &\leqslant \frac{4}{3} \left( 2^{1-2^k} + 2^{1-2^{k+1}} \right) d_{\mathbb{P}}(z, \zeta) \hat{\gamma}_{\mathrm{Frob}}(F, \zeta) \end{align} $$

(4.32) $$ \begin{align} &\leqslant \frac{1}{10} 2^{-2^k}. \end{align} $$

Besides, by Theorem 3.3, we have with probability at least $\frac 34$ ,

(4.33) $$ \begin{align} c \leqslant 192 n^2{D} \cdot \hat{\gamma}_{\mathrm{Frob}}(F, z^*) \end{align} $$

(note that the computation of c involves n calls to GammaProb, each returning a result outside the specified range with probability at most $\frac {1}{4n}$ . So the n computations are correct with probability at least $\frac 34$ ). It follows from (4.32) and (4.33), along with the choice of k, that

(4.34) $$ \begin{align} 2^{2^{k-1}} \beta(F,z^*) c \leqslant \tfrac{192}{10} n^2 {D} 2^{-2^{k-1}} \leqslant \alpha_0, \end{align} $$

with probability at least $\frac 34$ . We conclude, assuming (4.26), that Algorithm BlackBoxSolve succeeds with probability at least $\frac 34$ .

Assume now that the algorithm succeeds, but $z^*$ , the output point, is not an approximate zero of F. On the one hand, $z^*$ is not an approximate zero, so

(4.35) $$ \begin{align} \beta(F, z^*) \hat{\gamma}_{\mathrm{Frob}}(F, z^*)> \alpha_0, \end{align} $$

and on the other hand, the algorithm succeeds, so $2^{2^{k-1}} \beta (F, z^*) c \leqslant \alpha _0$ , and then

(4.36) $$ \begin{align} 2^{2^{k-1}} c \leqslant \hat{\gamma}_{\mathrm{Frob}}(F, z^*). \end{align} $$

By definition (2.16) of $\hat {\gamma }_{\mathrm {Frob}}(F, z^*)$ , and since $c = \kappa (F,z) (\Gamma _1^2 + \cdots +\Gamma _n^2)^{\frac 12}$ , where $\Gamma _i$ denotes the value returned by the call to , we get

(4.37) $$ \begin{align} 2^{2^{k-1}} (\Gamma_1^2 + \cdots +\Gamma_n^2)^{\frac12} \leqslant \left( \gamma_{\mathrm{Frob}}(f_1,z^*)^2+\dotsb+\gamma_{\mathrm{Frob}}(f_n, z^*)^2 \right)^{\frac12}. \end{align} $$

This implies that, for some i,

(4.38) $$ \begin{align} 2^{2^{k-1}} \Gamma_i \leqslant \hat{\gamma}_{\mathrm{Frob}}(f_i, z^*). \end{align} $$

By the choice of k, $2^{2^{k-1}} \geqslant \varepsilon ^{-1}$ , and using the tail bound in Theorem 3.3, with $t=\varepsilon ^{-1}$ , (4.38) may only happen with probability at most

(4.39) $$ \begin{align} \frac{1}{4n}\Big(\frac{1}{4n}\Big)^{\frac12 \log_2 t} \le \left(\frac{1}{4}\right)^{\frac12 \log_2 t} = 2^{- \log_2 t} = \varepsilon. \end{align} $$

The complexity bound is clear, since a Newton’s iteration requires only $\operatorname {poly}(n, {D}) L(F)$ operations.

The combination of BlackBoxSolve and Boost leads to Algorithm 7, BoostBlackBoxSolve.

Proof of Corollary 1.3

Let ${\mathbf u}\in {\mathcal {U}}$ be uniformly distributed and independent from F. By hypothesis, ${\mathbf u}\cdot F$ and F have the same distribution, so we study ${\mathbf u}\cdot F$ instead. Then Theorem 1.2 applies, and we obtain, for fixed $F\in {\mathcal {H}}$ and random $u\in {\mathcal {U}}$ , that BoostBlackBoxSolve terminates after

(4.40) $$ \begin{align} \operatorname{poly}(n, {D}) \cdot L \cdot \left( {\mathbb{E}} \left[ \Gamma(F) \log \Gamma(F) \right] + \log \log \varepsilon ^{-1} \right) \end{align} $$

operations on average. With the concavity on $[1,\infty )$ of the function $h:x\mapsto x^{\frac 12} \log x^{\frac 12}$ , Jensen’s inequality ensures that

(4.41) $$ \begin{align} {\mathbb{E}} \left[ \Gamma(F)\log \Gamma(F) \right] = {\mathbb{E}} \left[ h(\Gamma(F)^2) \right] \leqslant h \left( {\mathbb{E}}[\Gamma(F)^2] \right), \end{align} $$

which gives the complexity bound.

5.1 A coarea formula

The goal of this subsection is to establish a consequence of the coarea formula [Reference Federer19, Theorem 3.1] that is especially useful to estimate $\Gamma (f)$ for a random polynomial f. This involves a certain identity of normal Jacobians of projections that appears so frequently that it is worthwhile to provide the statement in some generality. Let us first introduce some useful notations. For a linear map $h : E \to F$ between two real Euclidean spaces, we define its Euclidean determinant as

(5.6) $$ \begin{align} \operatorname{Edet}(h) \,{\stackrel{.}{=}} \,\sqrt{{\operatorname{det}\nolimits}_{\mathbb R}(h\circ h^*)} , \end{align} $$

where $h^* : F\to E$ is the adjoint operator of h and the notation ${\operatorname {det}\nolimits }_{\mathbb {R}}$ simply recalls that the base field is $\mathbb {R}$ .

If $p : U\to V$ is a linear map between complex Hermitian spaces, then $\operatorname {Edet}(p)$ is defined by the induced real Euclidean structures on U and V, and it is well known that

(5.7) $$ \begin{align} \operatorname{Edet}(p) = {\operatorname{det}\nolimits}_{\mathbb{C}}(p \circ p^*), \end{align} $$

and ${\operatorname {det}\nolimits }_{\mathbb {C}}$ is the determinant over $\mathbb {C}$ .

The normal Jacobian of a smooth map $\varphi $ between Riemannian manifolds at a given point x is defined as the Euclidean determinant of the derivative of the map at that point:

(5.8) $$ \begin{align} \operatorname{NJ}_x \varphi \,{\stackrel{.}{=}}\, \operatorname{Edet} \left( \mathrm{d} _x \varphi \right). \end{align} $$

Proof. By symmetry, it suffices to show the first equality. Let $v_1,\dotsc ,v_r, w_1,\dotsc ,w_s$ be an orthonormal basis of $E\times F$ , such that $v_1,\dotsc ,v_r$ is a basis of V and $w_1,\dotsc ,w_s$ is a basis of $V^\perp $ . After fixing orthonormal bases for E and F (and the corresponding basis of $E\times F$ ), consider the orthogonal (respectively, unitary) matrix U with columns $v_1,\dotsc ,v_r, w_1,\dotsc ,w_s$ . We decompose U as a block matrix

(5.9)

Using $U U^*=I$ and $U^* U = I$ , we see that $V_E^{\phantom {*}} V_E^* + W_E^{\phantom {*}} W_E^* = I$ and $W^*_E W^{\phantom {*}}_E + W_F^* W_F^{\phantom {*}} = I$ . It follows from Sylvester’s determinant identity ${\operatorname {det}\nolimits }(I+AB)={\operatorname {det}\nolimits }(I+BA)$ that

(5.10) $$ \begin{align} {\operatorname{det}\nolimits}(V_E^{\phantom{*}} V_E^*) = {\operatorname{det}\nolimits}( I- W_E^{\phantom{*}} W_E^*) = {\operatorname{det}\nolimits}( I- W_E^* W_E^{\phantom{*}} ) = {\operatorname{det}\nolimits}(W_F^* W_F^{\phantom{*}}). \end{align} $$

By definition, we have $\operatorname {Edet}(p|_V) = {\operatorname {det}\nolimits }(V_E^{\phantom {*}} V^*_E )^{\frac {\eta }{2}}$ with $\eta =1$ in the Euclidean situation and $\eta =2$ in the Hermitian situation. Similarly, $\operatorname {Edet}((q|_{V^\perp })^*) = {\operatorname {det}\nolimits }(W_F^* W_F^{\phantom {*}})^{\frac \eta 2}$ . Therefore, we have $\operatorname {Edet}(p|_V) = \operatorname {Edet}((q|_{V^\perp })^*)$ .

Corollary 5.5. In the setting of Lemma 5.4, suppose V is a real (or complex) hyperplane in $E\times F$ with nonzero normal vector $(v,w)\in E \times F$ . Then

$$\begin{align*}\frac{\operatorname{Edet}(p|_{V})}{\operatorname{Edet}(q|_{V})} = \left(\frac{\|w\|}{\|v\|}\right)^\eta, \end{align*}$$

where $\eta =1$ in the Euclidean situation and $\eta =2$ in the Hermitian situation.

We consider now the abstract setting of a family $(f_A)$ of homogeneous polynomials of degree ${D}$ in the variables $z_0,\dotsc ,z_n$ , parameterised by elements A of a Hermitian manifold M through a holomorphic map $A \in M\mapsto f_A$ . Let ${\mathcal {V}}$ be the solution variety $\left \{ (A,\zeta )\in M\times {{\mathbb {P}}^n} \mathrel {}\middle |\mathrel {} f_A(\zeta ) = 0 \right \}$ and $\pi _1 : {\mathcal {V}}\to M$ and $\pi _2 : {\mathcal {V}} \to {\mathbb {P}}^n$ be the restrictions of the canonical projections. We can identify the fibre $\pi _1^{-1}(A)$ with the zero set $V(f_A)$ in ${{\mathbb {P}}^n}$ . Moreover, the fibre $\pi _2^{-1}(\zeta )$ can be identified with $M_{\zeta } \,{\stackrel {.}{=}}\, \left \{ A\in M\mathrel {}\middle |\mathrel {} f_A(\zeta ) = 0 \right \}$ . For fixed $\zeta \in {\mathbb {C}}^{n+1}$ , such that $\|\zeta \|=1$ , we consider the map $M\to {\mathbb {C}},\, A \mapsto f_A(\zeta )$ and its derivative at A,

(5.11) $$ \begin{align} \partial_A f(\zeta) : T_A M \to {\mathbb{C}}. \end{align} $$

Moreover, for fixed $A\in M$ , we consider the map $f_A\colon {\mathbb {C}}^{n+1}\to {\mathbb {C}}$ and its derivative at $\zeta $ ,

(5.12) $$ \begin{align} \mathrm{d} _{\zeta} f_A : T_{\zeta}{{\mathbb{P}}^n} \to {\mathbb{C}} , \end{align} $$

restricted to the tangent space $T_{\zeta }{{\mathbb {P}}^n}$ , that we identify with the orthogonal complement of ${\mathbb {C}}\zeta $ in ${\mathbb {C}}^{n+1}$ with respect to the standard Hermitian inner product. Below, it will be convenient to write $\partial _A f(\zeta )$ and $\mathrm {d} _{\zeta } f_A$ for $\zeta \in \mathbb {P}^n$ , meaning that $\zeta $ stands for a representative in $\mathbb {C}^{n+1}$ (due to $\partial _A f(t \zeta ) = t^{D} \partial _A f(\zeta )$ and $\mathrm {d} _{t\zeta } f_A = t^{D} \mathrm {d} _{\zeta } f_A$ , the statements do not depend on the scaling).

Proposition 5.6. For any measurable function $\Theta : {\mathcal {V}}\to [0,\infty )$ , we have

Here, $\mathrm {d} A$ denotes the Riemannian volume measure on M and $M_{\zeta }$ , respectively.

Proof. As in [Reference Bürgisser and Cucker15, Lemma 16.9], the tangent space of ${\mathcal {V}}$ at $(A,\zeta ) \in {\mathcal {V}}$ can be expressed as

(5.13) $$ \begin{align} V \,{\stackrel{.}{=}}\, T_{A,\zeta} {\mathcal{V}} = \left\{ (\dot A, \dot\zeta)\in T_A M\times T_{\zeta}{{\mathbb{P}}^n} \mathrel{}\middle|\mathrel{} \mathrm{d} _{\zeta} f_A(\dot\zeta) + \partial_A f(\zeta)(\dot A) = 0 \right\}. \end{align} $$

If $\partial _A f(\zeta )$ and $\mathrm {d} _{\zeta } f_A$ are not both zero, then V is a hyperplane in the product $E\times F \,{\stackrel {.}{=}}\, T_A M \times T_{\zeta }{{\mathbb {P}}^n}$ of Hermitian spaces and V has the normal vector $((\partial _A f(\zeta ))^*, (\mathrm {d} _{\zeta } f_A)^*)$ , where ${}^*$ denotes the Hermitian adjoint. If we denote by p and q the canonical projections of V onto E and F, then $\mathrm {d} _{A,\zeta } \pi _1 = p|_V$ and $\mathrm {d} _{A,\zeta } \pi _2 = q|_V $ , hence

(5.14) $$ \begin{align} \operatorname{NJ}_{A,\zeta}(\pi_1) = \operatorname{Edet}(p|_V), \quad \operatorname{NJ}_{A,\zeta}(\pi_2) = \operatorname{Edet}(q|_V). \end{align} $$

By Corollary 5.5, we, therefore, have

(5.15) $$ \begin{align} \frac{\operatorname{NJ}_{A,\zeta}(\pi_1)}{\operatorname{NJ}_{A,\zeta}(\pi_2)} = \frac{\operatorname{Edet}(p|_V)}{\operatorname{Edet}(q|_V)} = \frac{\|(\mathrm{d} _{\zeta} f_A)^*\|^2}{\|(\partial_A f(\zeta))^*\|^2} = \frac{\|\mathrm{d} _{\zeta} f_A\|^2}{\|\partial_A f(\zeta)\|^2}. \end{align} $$

The coarea formula [Reference Federer19, Theorem 3.1], applied to $\pi _1 : \mathcal {V}\to M$ asserts,

(5.16)

(note that ${\mathcal {V}}$ may have singularities, so we actually apply the coarea formula to its smooth locus. The Lipschitz-continuity hypothesis required by Federer is satisfied: $\pi _1$ is 1-Lipschitz continuous, since it is the restriction to ${\mathcal {V}}$ of the projection $M\times \mathbb {P}^n\to M$ ). On the other hand, the coarea formula applied to $\pi _2:{\mathcal {V}} \to \mathbb {P}^n$ gives

(5.17)

By (5.15), we have

(5.18) $$ \begin{align} \operatorname{NJ}_{A,\zeta}(p) \|\partial_A f(\zeta)\|^2 = \operatorname{NJ}_{A,\zeta}(q) \|\mathrm{d} _{\zeta} f_A\|^2 , \end{align} $$

so all the four integrals above are equal.

5.2 A few lemmas on Gaussian random matrices

We present, here, some auxiliary results on Gaussian random matrices, centring around the new notion of the anomaly of a matrix. This will be crucial for the proof of Theorem 1.5.

We endow the space $\mathbb {C}^r$ with the probability density $\pi ^{-r} e^{-\|x\|^2} \mathrm {d} x$ , where $\|x\|$ is the usual Hermitian norm, and call a random vector $x\in \mathbb {C}^r$ with this probability distribution standard Gaussian. This amounts to say that the real and imaginary parts of x are independent centred Gaussian with variance $\frac 12$ . We note that ${\mathbb {E}}_x \left [\|x\|^2\right ] = r$ . This convention slightly differs from some previous writings with a different scaling, where the distribution used is $(2\pi )^{-r} e^{-\frac 12\|x\|^2} \mathrm {d} x$ . This choice seems more natural since it avoids many spurious factors. Similarly, the matrix space $\mathbb {C}^{r\times s}$ is endowed with the probability density $\pi ^{-rs} \exp (-\|R\|_{\mathrm {Frob}}^2) \mathrm {d} R$ , and we call a random matrix with this probability distribution standard Gaussian as well (in the random matrix literature, this is called complex Ginibre ensemble).

Lemma 5.7. For $P\in \mathbb {C}^{r\times s}$ fixed and $x \in \mathbb {C}^s$ standard Gaussian, we have

$$ \begin{align*}{\mathbb{E}}_x \left[\|P x\|^2\right] = \|P\|_{\mathrm{Frob}}^2 , \quad {\mathbb{E}}_x \left[\|P x\|^{-2}\right] \ge\|P\|_{\mathrm{Frob}}^{-2} ,\quad {\mathbb{E}}_x \left[\|x\|^{-2}\right] = \frac{1}{s-1}. \end{align*} $$

We define the anomaly of a matrix $P \in \mathbb {C}^{r\times s}$ as the quantity

(5.19) $$ \begin{align} \theta(P) \,{\stackrel{.}{=}}\, {\mathbb{E}}_x \left[ \frac{\|P\|_{\mathrm{Frob}}^2}{\|P x\|^2} \right] \in [1,\infty) , \end{align} $$

where $x \in \mathbb {C}^s$ is a standard Gaussian random vector. Note that $\theta (P) \ge 1$ by Lemma 5.7. Moreover, by the same lemma, $\theta (I_r) = r/(r-1)$ . This quantity $\theta (P)$ is easily seen to be finite if $\operatorname {rk} P> 1$ ; it grows logarithmically to infinity as P approaches a rank 1 matrix.

Lemma 5.8. Let $P \in \mathbb {C}^{r\times s}$ and $Q \in \mathbb {C}^{t\times u}$ be fixed matrices and $X \in {\mathbb {C}^{s\times t}}$ be a standard Gaussian random matrix. Then

$$\begin{align*}{\mathbb{E}}_X \left[ \frac{\|P\|_{\mathrm{Frob}}^2 \|Q\|_{\mathrm{Frob}}^2}{\|P X Q\|_{\mathrm{Frob}}^2} \right] \leqslant \theta(P). \end{align*}$$

Proof. Up to left and right multiplications of Q by unitary matrices, we may assume that Q is diagonal, with nonnegative real numbers $\sigma _1,\dotsc ,\sigma _{\min (t, u)}$ on the diagonal (and we define $\sigma _ i=0$ for $i> \min (t, u)$ ). This does not change the left-hand side because the Frobenius norm is invariant by left and right multiplications with unitary matrices, and the distribution of X is unitarily invariant as well.

Let $e^u_1,\dotsc ,e^u_u$ (reps. $e_1^t, \dotsc , e_t^t)$ be the canonical basis of $\mathbb {C}^u$ (respectively, $\mathbb {C}^t$ ). Observe that

(5.20) $$ \begin{align} \|PXQ\|_{\mathrm{Frob}}^2 = \sum_{i=1}^u \|PXQ e^u_i\|^2 = \sum_{i=1}^{t} \sigma_i^2 \|PX e^{t}_i \|^2. \end{align} $$

Noting that $\|Q\|_{\mathrm {Frob}}^2 = \sigma _1^2 + \dotsb + \sigma _{t}^2$ , the convexity of $x \mapsto x^{-1}$ on $(0,\infty )$ gives

(5.21) $$ \begin{align} \left( \frac{1}{\|Q\|_{\mathrm{Frob}}^2} \sum_{i=1}^{t} \sigma_i^2 \|P X e^{t}_i \|^2 \right)^{-1} \leqslant \frac{1}{\|Q\|_{\mathrm{Frob}}^2} \sum_{i=1}^{t} \frac{\sigma_i^2}{\|P Xe^{t}_i\|^2}. \end{align} $$

Since X is standard Gaussian, $X e^{t}_i \in \mathbb {C}^s$ is also standard Gaussian. Therefore, by definition of $\theta $ , we have for any $1\leqslant i\leqslant t$ ,

(5.22) $$ \begin{align} {\mathbb{E}}_X \left[ \frac{\|P\|_{\mathrm{Frob}}^2}{\|P X e^{t}_i\|^2} \right] = \theta(P). \end{align} $$

It follows that

$$ \begin{align*} {\mathbb{E}}_X \left[ \frac{\|P\|_{\mathrm{Frob}}^2 \|Q\|_{\mathrm{Frob}}^2}{\|P X Q\|_{\mathrm{Frob}}^2} \right] &\leqslant \frac{1}{\|Q\|_{\mathrm{Frob}}^2} \sum_{i=1}^t \sigma_i^2 {\mathbb{E}} \left[ \frac{\|P\|_{\mathrm{Frob}}^2}{\|P Xe^{t}_i\|^2} \right], && \text{by~(5.20) and (5.21),}\\ &= \frac{1}{\|Q\|_{\mathrm{Frob}}^2} \sum_{i=1}^t \sigma_i^2 \theta(P), &&\text{by (5.22),} \\ & = \theta(P), \end{align*} $$

which concludes the proof.

Lemma 5.9. Let $P \in \mathbb {C}^{r\times s}$ be fixed, $t>1$ and $X\in \mathbb {C}^{s\times t}$ be a standard Gaussian random matrix. Then

$$\begin{align*}{\mathbb{E}}_X \left[ \theta(PX) \right] = \frac{1}{t-1} + \theta(P). \end{align*}$$

Furthermore, if $X_1,\dotsc ,X_m$ are standard Gaussian matrices of size $r_{0}\times r_1$ , $r_1\times r_2, \ldots , r_{m-1}\times r_m$ , respectively, where $r_0,\ldots ,r_m>1$ , then

$$\begin{align*}{\mathbb{E}}_{X_1,\ldots,X_m} \left[ \theta(X_1 \dotsb X_m ) \right] = 1 + \sum_{i=0}^m \frac{1}{r_i-1}. \end{align*}$$

Proof. Let $x \in \mathbb {C}^t$ be a standard Gaussian random vector, so that

(5.23) $$ \begin{align} {\mathbb{E}}_X \left[ \theta(PX) \right] = {\mathbb{E}}_{X,x} \left[ \frac{\|P X\|_{\mathrm{Frob}}^2}{\|P X x\|^2} \right]. \end{align} $$

We first compute the expectation conditionally on x. So we fix x and write $x = \|x\| u_1$ for some unit vector $u_1$ . We choose other unit vectors $u_2,\dotsc ,u_t$ to form an orthonormal basis of $\mathbb {C}^t$ . Since $\|PX\|_{\mathrm {Frob}}^2 = \sum _{i=1}^t \|PXu_i\|^2$ , we obtain

(5.24) $$ \begin{align} \frac{\|PX\|_{\mathrm{Frob}}^2}{\|PXx\|^2} = \frac{1}{\|x\|^2} + \sum_{i=2}^t \frac{\|PX u_i\|^2}{\|PXu_1\|^2 \|x\|^2}. \end{align} $$

Since X is standard Gaussian, the vectors $X u_i$ are standard Gaussian and independent. So we obtain, using Lemma 5.7,

(5.25) $$ \begin{align} {\mathbb{E}}_X \left[ \frac{\|PX u_i\|^2}{\|PXu_1\|^2} \right] &= {\mathbb{E}}_X [ \|PXu_i\|^2 ] \, {\mathbb{E}}_X \left[ \frac{1}{\|PX u_1\|^2} \right] \end{align} $$

(5.26) $$ \begin{align} &= \|P\|_{\mathrm{Frob}}^2 \, {\mathbb{E}}_X \left[ \frac{1}{\|PX u_1\|^2} \right] = \theta(P). \end{align} $$

Combining with (5.24), we obtain

(5.27) $$ \begin{align} {\mathbb{E}}_{X} \left[ \frac{\|P X\|_{\mathrm{Frob}}^2}{\|P X x\|^2} \right] = \frac{1}{\|x\|^2} + \sum_{i=2}^t \frac{\theta(P)}{\|x\|^2} = \frac{1}{\|x\|^2} \big(1+ (t-1) \theta(P)\big). \end{align} $$

When we take the expectation over x, the first claim follows with the third statement of Lemma 5.7.

The second claim follows by induction on m. The base case $m=1$ follows from writing ${\mathbb {E}}_{X_1} \left [ \theta (X_1) \right ]={\mathbb {E}}_{X_1} \left [ \theta (I_{r_0}X_1) \right ]$ , the first part of Lemma 5.7 and $\theta ( I_{r_0} ) = 1+\frac {1}{r_0-1}$ . For the induction step $m>1$ , we first fix $X_1,\ldots ,X_{m-1}$ and obtain from the first assertion

(5.28) $$ \begin{align} {\mathbb{E}}_{X_{m}} \left[ \theta(X_1 \dotsb X_{m-1} X_m ) \right] =\frac{1}{r_m-1} + \theta(X_1 \dotsb X_{m-1}). \end{align} $$

Taking the expectation over $X_1,\ldots ,\dotsb , X_{m-1}$ and applying the induction hypothesis implies the claim.

5.3 Proof of Proposition 5.3

We now carry out the estimation of

(5.29) $$ \begin{align} {\mathbb{E}} \left[ \left\| \mathrm{d} _{\zeta} f_{A} \right\|^{-2} \left\|{\tfrac{1}{k!}}\mathrm{d} _{\zeta}^{k}f_{A}\right\|_{\mathrm{Frob}}^{2} \right], \end{align} $$

where $A \in M_{\mathbf r}$ is standard Gaussian and $\zeta \in \mathbb {P}^n$ is a uniformly distributed zero of $f_{A}$ . The computation is lengthy, but the different ingredients arrange elegantly.