A Numerical Algorithm for Zero Counting. II: Distance to Ill-posedness and Smoothed Analysis

TL;DR

This paper establishes a condition number theorem linking zero counting for real polynomial systems to the inverse of the normalized distance to ill-posed systems, enabling smoothed analysis of the problem.

Contribution

It introduces a condition number theorem for zero counting, connecting it to the distance to ill-posed systems, and applies smoothed analysis to this condition number.

Findings

Condition number equals inverse of normalized distance to ill-posed systems

Smoothed analysis of the condition number is derived

Provides theoretical foundation for numerical stability in zero counting

Abstract

We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows.

Equations

$$\frac{\tilde{\kappa }(f)}{\sqrt{n}}\le \kappa (f)\le \sqrt{2n}\tilde{\kappa }(f).$$

$$B(a,\sigma ):=\{x\in {\mathbb{P}}^p(\mathbb{R})\mid d_{\mathbb{P}}(x,a)\le \sigma \}.$$

$$\mathcal{C}(z)=\frac{1}{d_{\mathbb{P}}(z,S)}.$$

$$\underset{a\in {\mathbb{P}}^p}{\sup }{\mathrm{Prob}}_{z\in B(a,\sigma )}\{\mathcal{C}(z)\ge t\}\le 13dp\frac{1}{\sigma t}$$

$$\underset{a\in {\mathbb{P}}^p}{\sup }{E}_{z\in B(a,\sigma )}[\ln \mathcal{C}(z)]\le 2\ln p+2\ln d+\ln \left(\frac{1}{\sigma }\right)+5.$$

$$\underset{f\in \mathbb{P}({\mathcal{H}}_{\mathbf{d}})}{\sup }{\mathrm{Prob}}_{g\in B(f,\sigma )}\{\tilde{\kappa }(g)\ge t\}\le 13n^2\mathbf{D}\mathcal{D}N\frac{1}{\sigma t}$$

$$\underset{f\in \mathbb{P}({\mathcal{H}}_{\mathbf{d}})}{\sup }{E}_{g\in B(f,\sigma )}[\ln \tilde{\kappa }(g)]\le 2\ln N+4\ln n+2\ln \mathcal{D}+\ln \mathbf{D}+\ln \left(\frac{1}{\sigma }\right)+6.$$

$${\mathrm{Prob}}_{g\in {\mathbb{P}}^p}\{\tilde{\kappa }(g)\ge t\}\le 13n^2\mathbf{D}\mathcal{D}N\frac{1}{t}$$

$$E_{g\in {\mathbb{P}}^p}[\ln \tilde{\kappa }(g)]\le 2\ln N+4\ln n+2\ln \mathcal{D}+\ln (\mathbf{D})+6.$$

$$⟨f,g{⟩}_W=\sum _{|j|=d}\frac{a_j{b}_j}{\left(\genfrac{}{}{0ex}{}{d}{j}\right)}$$

$$⟨f,g{⟩}_W=\sum _{i=1}^n⟨f_i,g_i{⟩}_W$$

$$\Sigma_{\mathbb{R}}(x)=\{f\in\mathcal{H}_{\mathbf{d}}\mid\mbox{$x$ is a multiple zero of $f$}\},$$

$$\Sigma_{\mathbb{R}}=\{f\in\mathcal{H}_{\mathbf{d}}\mid\mbox{$f$ has a multiple zero in $S^{n}$}\}=\bigcup_{x\in S^{n}}\Sigma_{\mathbb{R}}(x).$$

$${\tilde{\mu }}_{\mathrm{norm}}(f,x)=\Vert f{\Vert }_W\Vert Df(x{)}_{|T_x{S}^n}^{-1}\left[\begin{array}{cccc}\sqrt{d_1}& & & \\ & \sqrt{d_2}& & \\ & & \ddots & \\ & & & \sqrt{d_n}\end{array}\right]\Vert$$

$$\widetilde{\kappa}(f,x)=\frac{\|f\|_{W}}{\big{(}\|f\|_{W}^{2}\widetilde{\mu}_{\rm norm}(f,x)^{-2}+\|f(x)\|^{2}_{2}\big{)}^{1/2}}.$$

$$\tilde{\kappa }(f)=\underset{x\in S^n}{\max }\tilde{\kappa }(f,x)$$

$$\tilde{\kappa }(f,x)=\frac{1}{\mathsf{dist}(f,{\Sigma }_{\mathbb{R}}(x))}.$$

$$\mathsf{diag}\left(\frac{1}{\sqrt{d_{i}}}\right)Df(e_{0})_{|T_{e_{0}}S^{n}}=\underbrace{\left[\begin{array}[]{lll}&&\\ u_{1}&\ldots&u_{n}\\ &&\end{array}\right]}_{U}\left[\begin{array}[]{lll}\sigma_{1}&&\\ &\ddots&\\ &&\sigma_{n}\end{array}\right]V^{\rm t}$$

$$\Vert f_i-g_i{\Vert }_W^2={\left(\genfrac{}{}{0ex}{}{d_i}{d_i}\right)}^{-1}{f}_i(e_0{)}^2+{\left(\genfrac{}{}{0ex}{}{d_i}{d_i-1,1}\right)}^{-1}(\sqrt{d_i}{\sigma }_n{u}_{in}{)}^2=|f_i(e_0){|}^2+{\sigma }_n^2{u}_{in}^2$$

$$\Vert f-g{\Vert }_W^2=\Vert f(e_0){\Vert }_2^2+{\sigma }_n^2=\tilde{\kappa }(f,e_0{)}^{-2}.$$

$$\mathsf{dist}(f,{\Sigma }_{\mathbb{R}}(e_0))\le \Vert f-g{\Vert }_W=\tilde{\kappa }(f,e_0{)}^{-1}.$$

$$f_i(x)=f_i(e_0)x_0^{d_i}+\frac{\partial f_i}{\partial x_1}(e_0)x_0^{d_i-1}{x}_1+\cdots +\frac{\partial f_i}{\partial x_n}(e_0)x_0^{d_i-1}{x}_n+Q_i(x)$$

$$g_i(x)=\frac{\partial g_i}{\partial x_1}(e_0)x_0^{d_i-1}{x}_1+\cdots +\frac{\partial g_i}{\partial x_n}(e_0)x_0^{d_i-1}{x}_n+R_i(x).$$

$$\Vert f_i-g_i{\Vert }_W^2\ge |f_i(e_0){|}^2+\frac{1}{d_i}\Vert D{f}_i(e_0)-D{g}_i(e_0){\Vert }_2^2$$

$$\Vert f-g{\Vert }_W^2\ge \Vert f(e_0){\Vert }_2^2+{\Vert \mathsf{diag}\left(\frac{1}{\sqrt{d_i}}\right)Df(e_0)-\mathsf{diag}\left(\frac{1}{\sqrt{d_i}}\right)Dg(e_0)\Vert }_F^2.$$

$${\mathsf{dist}}_F\left(\mathsf{diag}\left(\frac{1}{\sqrt{d_i}}\right)Df(e_0),\mathsf{diag}\left(\frac{1}{\sqrt{d_i}}\right)Dg(e_0)\right)\ge {\mathsf{dist}}_F\left(\mathsf{diag}\left(\frac{1}{\sqrt{d_i}}\right)Df(e_0),{\mathsf{Sing}}_n\right)={\sigma }_n,$$

$$\Vert f-g{\Vert }_W^2\ge \Vert f(e_0){\Vert }_2^2+{\sigma }_n^2=\tilde{\kappa }(f,e_0{)}^{-2}.$$

$$\mathsf{dist}(f,{\Sigma }_{\mathbb{R}})=\underset{g\in {\Sigma }_{\mathbb{R}}}{\min }\mathsf{dist}(f,g)=\underset{x\in S^n}{\min }\mathsf{dist}(f,{\Sigma }_{\mathbb{R}}(x))$$

$$\tilde{\kappa }(f)=\underset{x\in S^n}{\max }\tilde{\kappa }(f,x)=\underset{x\in S^n}{\max }\frac{1}{\mathsf{dist}(f,{\Sigma }_{\mathbb{R}}(x))}=\frac{1}{\underset{x\in S^n}{\min }\mathsf{dist}(f,{\Sigma }_{\mathbb{R}}(x))}=\frac{1}{\mathsf{dist}(f,{\Sigma }_{\mathbb{R}})}.$$

$$\mathrm{deg}({\Sigma }_{\mathbb{C}})=n\mathcal{D}+(d_1+\cdots +d_n-n-1)\mathcal{D}\sum _{j=1}^n\frac{1}{d_j}.$$

Full text

A Numerical Algorithm for Zero Counting.

II: Distance to Ill-posedness and Smoothed Analysis

Felipe Cucker

Dept. of Mathematics

City University of Hong Kong

HONG KONG

e-mail: macucker@cityu.edu.hk Partially supported by City University SRG grant 7002106.   

Teresa Krick

Departamento de Matemática

Univ. de Buenos Aires & CONICET

ARGENTINA

e-mail: krick@dm.uba.ar Partially supported by grants ANPCyT 33671/05 and UBACyT X113/08-10.   

Gregorio Malajovich

Depto. de Matemática Aplicada

Univ. Federal do Rio de Janeiro

BRASIL

e-mail: gregorio@ufrj.br Partially supported by CNPq grants 470031/2007-7, 303565/2007-1 and by FAPERJ grant E26/110.849/2009.   

Mario Wschebor

Centro de Matemática

Universidad de la República

URUGUAY

e-mail: wschebor@cmat.edu.uy

Abstract. We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows.

1 Introduction

This paper continues the work in [8], where we described a numerical algorithm to count the number of zeros in $n$-dimensional real projective space of a system of $n$ real homogeneous polynomials. The algorithm works with finite precision and both its complexity and the precision required to ensure correctness are bounded in terms of $n$, the maximum ${\mathbf{D}}$ of the polynomials’ degrees, and a condition number $\kappa(f)$.

In this paper we replace $\kappa(f)$ —which was originally defined using the computationally friendly infinity norm— for a version $\widetilde{\kappa}(f)$ (defined in Section 2 below) which uses instead Euclidean norms. This difference is of little consequence in complexity estimates since one has (cf. Proposition 3.3 below) that

[TABLE]

It allows one, however, to prove a result following a classical theme in conditioning —the relation between condition and distance to ill-posedness— and to deduce from this result a probabilistic analysis for $\widetilde{\kappa}(f)$ and, a fortiori, for the complexity of the algorithm in [8]. This analysis is of interest since, in contrast with $n$ and ${\mathbf{D}}$, the value of $\widetilde{\kappa}(f)$ (or of $\kappa(f)$) is not apparent in $f$ and therefore complexity or accuracy bounds depending on this condition number are not of immediate use. A solution pioneered by John von Neumann and collaborators (see [14, §2.1] and references therein) and reintroduced by Steve Smale [23, 24] is to assume a probability measure on the space of data and to study the condition number at data $f$ as a random variable. This approach yields bounds on accuracy or complexity for random data and has been pursued in several contexts: systems of linear equations [12, 13], polyhedral conic systems [3, 5, 9, 15, 25], linear programs [4], complex polynomial systems [21], [18], etc. In our case it allows us to trade the presence of $\widetilde{\kappa}(f)$ in deterministic bounds for probabilistic bounds in $n$ and ${\mathbf{D}}$ only.

1.1 Distance to Ill-posedness

It has since long been observed [11] that the condition number for several problems (in its original aception, as a measure of the worst possible magnification of small input errors in the output [19]) either coincides with the relativized inverse of the distance from the input to the set of ill-posed data or is bounded by a small multiple of this inverse. A data is ill-posed when the magnification above is unbounded. In our case, a polynomial system $f$ is ill-posed when arbitrary small perturbations of $f$ may change its number of projective real zeros. Systems having this property are exactly those having multiple (real projective) zeros. Let us denote the set of such systems by $\Sigma_{\mathbb{R}}$. Also, let $\mathcal{H}_{\mathbf{d}}$ denote our set of input systems, i.e. the set of $f=(f_{1},\dots,f_{n})$, $n$ real homogeneous polynomials in $n+1$ variables of degrees ${\bf d}:=(d_{1},\dots,d_{n})$ respectively, endowed with the Bombieri-Weyl norm $\|\ \|_{W}$ (defined in Section 2). Finally, let $\mathsf{dist}$ be the distance on $\mathcal{H}_{\mathbf{d}}$ induced by this norm.

Our main result in this note is the following.

Theorem 1.1

For all $f\in\mathcal{H}_{\mathbf{d}}$, $\widetilde{\kappa}(f)=\frac{\|f\|_{W}}{\mathsf{dist}(f,\Sigma_{\mathbb{R}})}$.

Remark 1.2

It is worth noting that, although $\widetilde{\kappa}(f)$ is somehow related to the condition number $\mu_{\rm norm}(f)$ for complex polynomial systems (cf. [1, 20]) a result like Theorem 1.1 does not hold for the latter. Actually, such a result holds on the fibers of the zeros (see [1, Ch. 12, Theorem 3]) but not globally.

We also note that, as a consequence of Theorem 1.1, we have $\widetilde{\kappa}(f)\geq 1$ for all $f\in\mathcal{H}_{\mathbf{d}}$. This feature, although not immediate from the definition of $\widetilde{\kappa}(f)$, follows immediately from the fact that $0\in\Sigma_{\mathbb{R}}$. **

1.2 Smoothed Analysis

Theorem 1.1 carries out meaningful consequences in the probabilistic analysis of $\widetilde{\kappa}(f)$.

The usual probabilistic analysis for polynomial systems assume that random $f$ are drawn from the unit sphere $S(\mathcal{H}_{\mathbf{d}})$ with the uniform distribution (or from such a distribution on the real projective space induced by this sphere). A different approach to the randomization of input data has been recently proposed under the name of smoothed analysis [22]. The idea consists on replacing ‘random data’ by ‘random perturbations of given data’. A recent result in [2] derives smoothed analysis bounds for condition numbers which can be written as inverses to distances to ill-posedness. Because of Theorem 1.1, these bounds can be straightforwardly applied to $\widetilde{\kappa}(f)$.

To state this result we need to introduce some notation. Let $\mathbb{P}^{p}(\mathbb{R})$ denote the real projective space of dimension $p$ and $d_{\mathbb{P}}$ be the projective distance on $\mathbb{P}^{p}(\mathbb{R})$ (i.e. the sinus of the Riemannian distance). For a point $a\in\mathbb{P}^{p}(\mathbb{R})$ and $\sigma\in(0,1]$ we denote by $B(a,\sigma)$ the ball (w.r.t. $d_{\mathbb{P}}$) centered at $a$ and of radius $\sigma$. That is,

[TABLE]

In what follows we assume $B(a,\sigma)$ endowed with the uniform probability measure. Note that in the particular case $\sigma=1$ we obtain $B(a,\sigma)=\mathbb{P}^{p}(\mathbb{R})$ for each $a\in\mathbb{P}^{p}$ and hence, the usual probabilsitic analysis referred to above.

Theorem 1.3

[2]* Let $S\subset\mathbb{P}^{p}$ be contained in a projective hypersurface $H$ of degree at most $d$ and ${\mathscr{C}}:\mathbb{P}^{p}\to[1,\infty]$ be given by*

[TABLE]

Then, for all $\sigma\in(0,1]$ and all $t\geq(2d+1)\frac{p}{\sigma}$,

[TABLE]

and

[TABLE]

As a consequence of this result we obtain the following corollary (which we prove in Section 3). Here $\mathbb{P}(\mathcal{H}_{\mathbf{d}})$ denotes the projective space associated to $\mathcal{H}_{\mathbf{d}}$, $N$ denotes its dimension and ${\mathcal{D}}=d_{1}\cdots d_{n}$ its associated Bézout number.

Corollary 1.4

For all $\sigma\in(0,1]$ and all $t\geq(4n{\mathcal{D}}^{2}+1)\frac{N}{\sigma}$,

[TABLE]

and

[TABLE]

In particular, taking $\sigma=1$, we obtain average analysis: for all $t\geq N(4n{\mathcal{D}}^{2}+1)$,

[TABLE]

and

[TABLE]

A recent result [7] extends Theorem 1.3 to absolutely continuous measures on $B(a,\sigma)$ whose densities are radially symmetric around $a$ and may have a pole at $a$. Applications of this result to $\widetilde{\kappa}(f)$ readily follow.

2 Setting and Notations

For $d\in\mathbb{N}$, $\mathcal{H}_{d}$ denotes the subspace of $\mathbb{R}[x_{0},\ldots,x_{n}]$ of homogeneous polynomials of degree $d$. We endow $\mathcal{H}_{d}$ with the Bombieri-Weyl inner product $\langle\ ,\ \rangle_{W}$, defined for $f=\sum_{|j|=d}a_{j}{\boldsymbol{x}}^{j}$ and $g=\sum_{|j|=d}b_{j}{\boldsymbol{x}}^{j}$ by

[TABLE]

where for $j=(j_{0},\dots,j_{n})$, ${d\choose j}:=\frac{d!}{j_{0}!\cdots j_{n}!}$. A main feature of this inner product is its invariance under the action of the orthogonal group $O(n+1)$. That is, for all $\psi\in O(n+1)$ and all $f,g\in\mathcal{H}_{d}$, $\langle f\circ\psi,g\circ\psi\rangle_{W}=\langle f,g\rangle_{W}$. Next, for $d_{1},\ldots,d_{n}\in\mathbb{N}$, we endow $\mathcal{H}_{\mathbf{d}}=\mathcal{H}_{d_{1}}\times\ldots\times\mathcal{H}_{d_{n}}$ with the inner product

[TABLE]

where $f=(f_{1},\dots,f_{n})$, $g=(g_{1},\dots,g_{n})\in\mathcal{H}_{\mathbf{d}}$. We write $\|\ \|_{W}$ and $\mathsf{dist}$ to denote the norm and distance on $\mathcal{H}_{\mathbf{d}}$ induced by this inner product.

Projective zeros of polynomial systems $f\in\mathcal{H}_{\mathbf{d}}$ correspond to pairs of zeros $(-\zeta,\zeta)$ of the restriction $f_{|S^{n}}$ of $f$ to the $n$-dimensional unit sphere $S^{n}\subset\mathbb{R}^{n+1}$. We will thus consider a system $f\in\mathcal{H}_{\mathbf{d}}$ as a (centrally symmetric) mapping of $S^{n}$ into $\mathbb{R}^{n}$.

For a point $x\in S^{n}$ and a system $f\in\mathcal{H}_{\mathbf{d}}$ one may define both ill-posedness and condition relative to this point. For the first, one defines

[TABLE]

the set of systems which are ill-posed at $x$. Note that $\Sigma_{\mathbb{R}}(x)\neq\emptyset$ for all $x\in S^{n}$ and that

[TABLE]

Towards the definition of $\widetilde{\kappa}(f)$, for $f\in\mathcal{H}_{\mathbf{d}}$ and $x\in S^{n}$, we define

[TABLE]

where $Df(x)_{|T_{x}S^{n}}$ is the restriction to the tangent space of $x$ at $S^{n}$ of the derivative of $f$ at $x$ and the norm is the spectral norm, i.e. the operator norm with respect to $\|\ \|_{2}$.

Next, we define the condition of $f$ relative to $x$ to be

[TABLE]

Finally, we take the condition number $\widetilde{\kappa}(f)$ of $f\in\mathcal{H}_{\mathbf{d}}$ to be its condition relative to its worst conditioned point,

[TABLE]

Note that for all $\lambda\neq 0$, $\widetilde{\kappa}(\lambda f)=\widetilde{\kappa}(f)$ and $\mathsf{dist}(\lambda f,\Sigma_{\mathbb{R}})=|\lambda|\mathsf{dist}(f,\Sigma_{\mathbb{R}})$. The same is true relative to a point $x\in S^{n}$. We will therefore assume, without loss of generality, that $\|f\|_{W}=1$, and denote by $S(\mathcal{H}_{\mathbf{d}})$ the unit sphere in $\mathcal{H}_{\mathbf{d}}$.

3 The proofs

3.1 The main results

Proposition 3.1

For all $x\in S^{n}$ and $f\in S(\mathcal{H}_{\mathbf{d}})$,

[TABLE]

Proof. For $0\leq i\leq n$, let $e_{i}=(0,\ldots,0,1,0,\ldots,0)$ denote the $i$th coordinate vector. The group $O(n+1)$ acts on $\mathcal{H}_{\mathbf{d}}\times S^{n}$ and leaves $\mu_{\rm norm},\widetilde{\kappa}$ and distance to $\Sigma_{\mathbb{R}}(\ )$ invariant. Therefore, we may assume without loss of generality that $x=e_{0}$. This implies that $T_{e_{0}}S^{n}\simeq\langle e_{1},\ldots,e_{n}\rangle$ and we may write the singular value decomposition

[TABLE]

with $U$ and $V$ orthogonal and $\sigma_{1}\geq\sigma_{2}\geq\ldots\geq\sigma_{n}\geq 0$. Since the subgroup of $O(n+1)$ leaving $e_{0}$ invariant is isomorphic to $O(n)$ acting on $T_{e_{0}}S^{n}$ we may as well assume that $V={\rm Id}$. Note that $\widetilde{\mu}_{\rm norm}(f,e_{0})=\sigma_{n}^{-1}$, and therefore $\widetilde{\kappa}(f,e_{0})=(\sigma_{n}^{2}+\|f(e_{0})\|_{2}^{2})^{-1/2}$.

Let $g_{i}(x):=f_{i}(x)-f_{i}(e_{0})x_{0}^{d_{i}}-\sqrt{d_{i}}\sigma_{n}u_{in}x_{0}^{d_{i}-1}x_{n}$, where $u_{n}=(u_{1n},\dots,u_{nn})^{\rm t}$. Clearly, $g_{i}(e_{0})=0$ and $Dg_{i}(e_{0})e_{n}=0$ (here and in the sequel we denote $Dg_{i}(e_{0})_{|T_{e_{0}}S^{n}}$ simply by $Dg_{i}(e_{0})$) since $\partial g_{i}/\partial x_{n}(e_{0})=\partial f_{i}/\partial x_{n}(e_{0})-\sqrt{d_{i}}u_{in}\sigma_{n}=0$. Thus, $g=(g_{1},\ldots,g_{n})\in\Sigma_{\mathbb{R}}(e_{0})$. Moreover

[TABLE]

and hence, using $\|u_{n}\|=1$,

[TABLE]

It follows that

[TABLE]

For the reciprocal, let $g\in\Sigma_{\mathbb{R}}(e_{0})$. Then, $g(e_{0})=0$ and $Dg(e_{0})$ is singular. We want to show that $\|f-g\|_{W}\geq\widetilde{\kappa}(f,e_{0})^{-1}$. To this end, we write

[TABLE]

with $\deg_{x_{0}}Q_{i}\leq d_{i}-2$ and, similarly,

[TABLE]

Then

[TABLE]

and

[TABLE]

We know that $\mathsf{diag}\left(\frac{1}{\sqrt{d_{i}}}\right)Dg(e_{0})$ is singular. Hence, denoting by $\mathsf{Sing}_{n}$ the set of singular $n\times n$ matrices and by $\mathsf{dist}_{F}$ the Frobenius distance on this set of matrices,

[TABLE]

the last by the Eckart-Young Theorem [1, §11.1]. It follows that

[TABLE]

Proof of Theorem 1.1.  Again we can assume $f\in S(\mathcal{H}_{\mathbf{d}})$. Note that

[TABLE]

since $\Sigma_{\mathbb{R}}=\bigcup_{x\in S^{n}}\Sigma_{\mathbb{R}}(x)$. Then,

[TABLE]

Before proving Corollary 1.4 we recall some useful facts in algebraic geometry.

For $1\leq i\leq n$, let $f_{i}=\sum_{|j|=d_{i}}u_{ij}{\boldsymbol{x}}^{j}$ be a generic (i.e. with indeterminate coefficients) homogeneous polynomial of degree $d_{i}$ in the variables ${\boldsymbol{x}}=(x_{0},\dots,x_{n})$ and $f=(f_{1},\dots,f_{n})$. Set $N:=\sum_{i=1}^{n}{d_{i}+n\choose n}-1$, the dimension of the projective coefficients space. The $\mathbf{d}$-discriminant variety $\Sigma_{\mathbb{C}}\subset\mathbb{P}^{N}(\mathbb{C})$ is the locus of such polynomial systems $f=(f_{1},\dots,f_{n})$ with multiple zeros, i.e. such that there exists $z\in\mathbb{C}^{n+1}$, $z\neq 0$, with $f_{1}(z)=\cdots=f_{n}(z)=0$ and $Df(z)$ has rank $<n$. It is well-known that $\Sigma_{\mathbb{C}}$ is a hypersurface in $\mathbb{P}^{N}(\mathbb{C})$ defined by an irreducible polynomial ${\rm Disc}(f)\in\mathbb{Z}[u_{ij}]$ (see [17] or [16, Ch. 10]). For lack of a precise reference we prove the following result.

Lemma 3.2

[TABLE]

Proof. We know that $\deg(\Sigma_{\mathbb{C}})=\deg({\rm Disc}(f))$. We apply Identity (13) of [10]:

[TABLE]

where the standard notation ${\rm Res}_{d_{1},\dots,d_{n}}$ is for the multihomogeneous projective resultant of $n$ generic homogeneous polynomials of respective degrees $d_{1},\dots,d_{n}$ in $n$ variables, $\rho:=d_{1}+\cdots+d_{n}-n$, $J_{f}$ is the determinant of the matrix $(\partial f_{i}/\partial x_{j})_{1\leq i,j\leq n}$ and $f_{i}^{0}$ denotes the homogeneous component (of degree $d_{i}$) of $f_{i}(1,x_{1},\dots,x_{n})$.

We note that $\deg(J_{f})$ is a polynomial of degree $\rho$ in ${\boldsymbol{x}}$ whose coefficients are polynomials in $u_{ij}$ of degree $n$. On the other hand ${\rm Res}_{d_{1},\dots,d_{n}}$ is a multihomogeneous polynomial of degree $\prod_{k\neq i}d_{k}$ in the group of variables $u_{ij}$ ([6]). Therefore, since $J_{f}$ has degree $n$ in the $u_{ij}$, we derive

[TABLE]

The statement easily follows. ∎

Proof of Corollary 1.4.  We have $\Sigma_{\mathbb{R}}\subset\Sigma_{\mathbb{C}}$ and for all $f\in\mathcal{H}_{\mathbf{d}}$, $f\neq 0$,

[TABLE]

The statement now follows from Theorem 1.3 with $p=N$, $S=\Sigma_{\mathbb{R}}\cap\mathbb{P}^{N}(\mathbb{R})$, $H=\Sigma_{\mathbb{C}}\cap\mathbb{P}^{N}(\mathbb{R})$, and $d=n^{2}{\mathbf{D}}{\mathcal{D}}$, the last by Lemma 3.2 since $n{\cal D}+(\rho-1){\cal D}\sum_{j=1}^{n}\frac{1}{d_{j}}\leq n^{2}{\mathbf{D}}\cal D$. ∎

3.2 The equivalence of condition numbers

In [8] we considered $\mathcal{H}_{\mathbf{d}}$ endowed with the norm given by

[TABLE]

(we note $\|f\|\leq\|f\|_{W}\leq\sqrt{n}\|f\|$) and defined

[TABLE]

with

[TABLE]

(we note $\widetilde{\mu}_{\rm norm}(f,x)\leq\mu_{\rm norm}(f,x)\leq\sqrt{n}\,\widetilde{\mu}_{\rm norm}(f,x)$). The next result shows that the $\kappa(f)$ thus defined is closely related to $\widetilde{\kappa}(f)$.

Proposition 3.3

[TABLE]

Proof. Let $x\in S^{n}$. We observe that since $\|f\|_{W}^{2}\widetilde{\mu}_{\rm norm}(f,x)^{-2}+\|f(x)\|_{2}^{2}\geq\|f\|_{W}^{2}\widetilde{\mu}_{\rm norm}(f,x)^{-2}$, we have $\widetilde{\kappa}(f,x)\leq\widetilde{\mu}_{\rm norm}(f,x)\leq\mu_{\rm norm}(f,x)$. Similarly, using $\|f\|_{W}\leq\sqrt{n}\|f\|$ and $\|f(x)\|_{2}\geq\|f(x)\|_{\infty}$, we have $\widetilde{\kappa}(f,x)\leq\sqrt{n}\|f\|/\|f(x)\|_{\infty}$. Therefore

[TABLE]

which implies

[TABLE]

To prove the other inequality note that, for any $x\in S^{n}$,

[TABLE]

and $\|f(x)\|_{2}\leq\sqrt{n}\,\|f(x)\|_{\infty}$. Therefore,

[TABLE]

This implies $\kappa(f)\leq\sqrt{2n}\,\widetilde{\kappa}(f)$. ∎

Acknowledgment. In June 2008 we discussed condition and distance to ill-posedness with Mike Shub who suggested to us that a result in the spirit of Theorem 1.1 should be true. We are grateful to him since that conversation is at the origin of our paper.