Probabilistic Condition Number Estimates For Real Polynomial Systems I: A Broader Family Of Distributions

TL;DR

This paper extends probabilistic estimates of the condition number for real polynomial systems, accommodating broader distributions and over-determined systems, with implications for sensitivity analysis and complexity.

Contribution

It introduces new probabilistic bounds for the condition number under wider measures and generalizes to over-determined systems, also providing Lipschitz estimates for polynomial maps.

Findings

Broader probabilistic estimates for condition numbers.

Extension to over-determined polynomial systems.

New Lipschitz bounds for polynomial maps.

Abstract

We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generalize further by allowing over-determined systems. Along the way, we derive new Lipshitz estimates for polynomial maps from R^n to R^m, extending earlier work of Kellog on the case m=1, which may be of independent interest. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which monomial terms can appear) can help further improve earlier condition number estimates.