Detecting rigid convexity of bivariate polynomials

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Given a polynomial x ∈ Rn {mapping} 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. © 2009 Elsevier Inc. All rights reserved.




Henrion, D. (2010). Detecting rigid convexity of bivariate polynomials. Linear Algebra and Its Applications, 432(5), 1218–1233.

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