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A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inclusion isotonicity of the related enclosure function is shown.

Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for range of p over V. Bounds for the range of a rational function over V can easily be obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds converge monotonically and linearly to the range of the rational function if the degree of the Bernstein expansion is elevated. If V is subdivided then the convergence is quadratic with respect to the maximum of the diameters of the subsimplices.

In 1970, B.A. Asner, Jr., proved that for a real quasi-stable polynomial, i.e., a polynomial whose zeros lie in the closed left half-plane of the complex plane, its finite Hurwitz matrix is totally nonnegative, i.e., all its minors are nonnegative, and that the converse statement is not true. In this work, we explain this phenomenon in detail, and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite Hurwitz matrix.