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Preprints, Working Papers, ... Year : 2022

Computing the set of asymptotic critical values of polynomial mappings from smooth algebraic sets

Jérémy Berthomieu
Andrew Ferguson
• Function : Author
• PersonId : 1076496
Mohab Safey El Din

Abstract

Let $\mathbf{f} = (f_1, \dots, f_p) \in \mathbb{Q}[z_1, \dots, z_n]$ be a polynomial tuple. Define the polynomial mapping $\mathbf{f}: X \to \mathbb{C}^p$, where $X$ is a smooth algebraic set defined by the simultaneous vanishing of the reduced regular sequence $g_1, \dots, g_m$, with $m+p \leq n$. Let $d = \max(\deg f_1, \dots, \deg f_p, \deg g_1, \dots, \deg g_m)$, $\operatorname{d} \mathbf{f}$ be the differential of $\mathbf{f}$ and $\kappa$ be the function measuring the distance of a linear operator to the set of singular linear operators from $\mathbb{C}^n$ to $\mathbb{C}^p$. We consider the problem of computing the set of asymptotic critical values of $\mathbf{f}$. This is the set of values $c$ in the target space of $\mathbf{f}$ such that there exists a sequence of points $(\mathbf{x}_i)_{i\in \mathbb{N}}$ tending to $\infty$ for which $\mathbf{f}(\mathbf{x}_i)$ tends to $c$ and $\|\mathbf{x}_i\|\kappa(\operatorname{d} \mathbf{f}(\mathbf{x}_i))$ tends to $0$ when $i$ tends to infinity. The union of the classical and asymptotic critical values contains the so-called bifurcation set of a polynomial mapping. Thus, by computing both the critical values and the asymptotic critical values, one can utilise generalisations of Ehresmann's fibration theorem in non-proper settings for applications in polynomial optimisation and computational real algebraic geometry. We design new efficient algorithms for computing the set of asymptotic critical values of a polynomial mapping restricted to a smooth algebraic set. By investigating the degree of the objects constructed in our algorithms, we give the first bound on the degree of this set of values of $pD$, where $D = d^{n-p-1} \sum_{i = 0}^{p+1} \binom{n+p-1}{m+2p+i} d^i$. We also give the first complexity analysis of this problem, showing that it requires at most $\tilde{O}\left( p (p+1) D^{p+5} + (n+m+2p)^{d+3} D^{p+4}\right)$ operations in the base field. Moreover, in the special case $p=1$, we give another complexity estimate of $\tilde{O}((n+m+2)^{d+3} D^5)$ arithmetic operations. Additionally, we show how to apply these algorithms to polynomial optimisation problems and the problem of computing sample points per connected component of a semi-algebraic set defined by a single inequality/inequation. We provide implementations of our algorithms and use them to test their practical capabilities. We show that our algorithms significantly outperform the current state-of-the-art algorithms by tackling previously out of reach benchmark examples.

Domains

Symbolic Computation [cs.SC]

Dates and versions

hal-03598352 , version 1 (04-03-2022)

Identifiers

• HAL Id : hal-03598352 , version 1

Cite

Jérémy Berthomieu, Andrew Ferguson, Mohab Safey El Din. Computing the set of asymptotic critical values of polynomial mappings from smooth algebraic sets. 2022. ⟨hal-03598352⟩

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