Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

• Authors: Sharon Hutton;Erich L. Kaltofen;Lihong Zhi
• Venue: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
• Year: 2010

We give a stability criterion for real polynomial inequalities with floating point or inexact scalars by estimating from below or computing the radius of semidefiniteness. That radius is the maximum deformation of the polynomial coefficient vector measured in a weighted Euclidean vector norm within which the inequality remains true. A large radius means that the inequalities may be considered numerically valid. The radius of positive (or negative) semidefiniteness is the distance to the nearest polynomial with a real root, which has been thoroughly studied before. We solve this problem by parameterized Lagrangian multipliers and Karush-Kuhn-Tucker conditions. Our algorithms can compute the radius for several simultaneous inequalities including possibly additional linear coefficient constraints. Our distance measure is the weighted Euclidean coefficient norm, but we also discuss several formulas for the weighted infinity and 1-norms. The computation of the nearest polynomial with a real root can be interpreted as a dual of Seidenberg's method that decides if a real hypersurface contains a real point. Sums-of-squares rational lower bound certificates for the radius of semidefiniteness provide a new approach to solving Seidenberg's problem, especially when the coefficients are numeric. They also offer a surprising alternative sum-of-squares proof for those polynomials that themselves cannot be represented by a polynomial sum-of-squares but that have a positive distance to the nearest indefinite polynomial.