Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n   variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4n=1,2,3,4, and we compare the computed range widths for random n  -variate polynomials for n⩽10n⩽10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

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► Comparison framework for range bound computation of polynomials based on polytopes.
► Convex hull of the coefficients from the tensorial Bernstein basis (exponential size).
► Scheme for Bernstein polytopes obtained from Bernstein polynomials (configurable size).
► Tight range bound computation by linear programming.