Detecting rigid convexity of bivariate polynomials

Didier Henrion (1)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

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 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.

Mots clés: real algebraic geometry; Polynomial; convexity; linear matrix inequality; real algebraic geometry.

Domaines: Optimisation et contrôle [math.OC]

Didier Henrion. Detecting rigid convexity of bivariate polynomials. Linear Algebra and its Applications, 2010, 432 (5), pp.1218-1233. ⟨10.1016/j.laa.2009.10.033⟩. ⟨hal-00214196⟩