Some interval Newton solvers rely on tensorial Bernstein bases to compute sharp enclosures of multivariate polynomials on the unit hypercube. These solvers compute all coefficients with respect to tensorial Bernstein bases. Unfortunately, polynomials become exponential size in tensorial Bernstein bases. This article gives the first polynomial time method to solve this issue. A polynomial number of relevant Bernstein polynomials is selected. The non-negativity of each of these Bernstein polynomials gives a linear inequality in a space connected to the monomials of the canonical tensorial basis. We resort to linear programming on the resulting Bernstein polytope to compute range bounds of a polynomial or bounds of the zero set.
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