Stable Complete Intersections

TL;DR

This paper investigates the stability of zeros of complete intersections under coefficient perturbations, providing methods to identify parameter regions with invariant zeros and techniques to enhance zero stability.

Contribution

It introduces a framework for analyzing zero stability in complete intersections and proposes modifications to improve robustness against coefficient perturbations.

Findings

Constructed semi-algebraic sets with invariant real zeros

Developed methods to modify complete intersections for increased zero stability

Provided theoretical insights into zero behavior under perturbations

Abstract

A complete intersection of n polynomials in n indeterminates has only a finite number of zeros. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which all the complete intersection ideals share the same number of isolated real zeros. In the second part we show how to modify the complete intersection and get a new one which generates the same ideal but whose real zeros are more stable with respect to perturbations of the coefficients.