Detecting rigid convexity of bivariate polynomials

Given a polynomial x ∈ Rn 7→ p(x) in n = 2 variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x : p(x) ≥ 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 C = {x : p(x) = 0} is an algebraic curve of genus zero, a second algorithm based on Bézoutians is proposed to detect whether P has an LMI representation and to build such a representation from a rational parametrization of C. Finally, some extensions to positive genus curves and to the case n > 2 are mentioned.