On semidefinite representations of plane quartics

Didier Henrion (1)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables. If the quartic is rationally parametrizable, an exact semidefinite representation with 2 lifting variables can be obtained. Various numerical examples illustrate the techniques and suggest further research directions.

Mots clés: semidefinite programming; polynomials; algebraic plane curves

Domaines: Optimisation et contrôle [math.OC]

Didier Henrion. On semidefinite representations of plane quartics. 2008. ⟨hal-00320368v2⟩