Detecting rigid convexity of bivariate polynomials
| dc.creator | Henrion, Didier | |
| dc.date | 2008-01-23 | |
| dc.date.accessioned | 2026-07-07T08:56:05Z | |
| dc.date.available | 2026-07-07T08:56:05Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/0801.3592 | |
| dc.identifier | http://arxiv.org/abs/0801.3592 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146484 | |
| dc.subject | Optimization and Control | |
| dc.title | Detecting rigid convexity of bivariate polynomials | |
| dc.type | text |