Detecting rigid convexity of bivariate polynomials

Given a polynomial x∈Rn↦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.