Real Stability Testing

TL;DR

This paper presents a strongly polynomial time algorithm for testing the real stability of bivariate polynomials, with implications for preserving real-rootedness under linear transformations, leveraging hyperbolic polynomial properties.

Contribution

It introduces a novel, efficient algorithm for real stability testing of bivariate polynomials, connecting hyperbolic polynomial properties to stability verification.

Findings

Algorithm runs in strongly polynomial time

Reduces stability testing to polynomial nonnegativity checks

Applicable to linear transformations preserving real-rootedness

Abstract

We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials preserves real-rootedness. The proof exploits properties of hyperbolic polynomials to reduce real stability testing to testing nonnegativity of a finite number of polynomials on an interval.