Detecting rigid convexity of bivariate polynomials

TL;DR

This paper presents symbolic-numerical algorithms to determine if a bivariate polynomial's sublevel set is rigidly convex and has an LMI representation, linking polynomial hyperbolicity to semidefinite programming.

Contribution

It introduces a novel algorithm for detecting rigid convexity of bivariate polynomials and constructing LMI representations using algebraic geometry tools.

Findings

Algorithm for checking positive semidefiniteness of polynomial matrices.

Method for constructing LMI representations from algebraic curve parametrizations.

Extensions discussed for higher genus curves and higher-dimensional cases.

Abstract

Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$ containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial $p(x)$ is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety ${\mathcal C} = \{x : p(x) = 0\}$ is an algebraic curve of genus zero, a second algorithm based on B\'ezoutians is proposed to detect whether $\mathcal P$ has an LMI representation and to build such a representation from a rational parametrization of $\mathcal C$. Finally, some extensions to positive genus curves and to the case $n>2$ are mentioned.