Detecting rigid convexity of bivariate polynomials

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 $\mathcal{P}=\{x:p(x)\ge 0\}$ 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 Bézoutians 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.