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 several polytopes based on the tensorial Bernstein basis, and we formulate a polytope for the quadratic patch Q _n:= (x₁, ..., x_n, x²₁, ..., x²_n, x₁x₂, ..., x _n-1^x_n) by projections. This Bernstein polytope has Θ(n²) hyperplanes. We give the number of vertices, the number of hyperplanes, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 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.