Bounds for the Range of a Bivariate Polynomial over a Triangle

Abstract

The problem of finding an enclosure for the range of a bivariate polynomial p over the unit triangle is considered. The polynomial p is expanded into Bernstein polynomials. If p has only real coefficients the coefficients of this expansion, the so-called Bernstein coefficients, provide lower and upper bounds for the range. In the case that p has complex coefficients the convex hull of the Bernstein coefficients encloses the range. The enclosure is improved by subdividing the unit triangle into squares and triangles and computing enclosures for the range of p over these regions. It is shown that the sequence of enclosures obtained in this way converges to the convex hull of the range in the Hausdorff distance. Furthermore, it is described how the Bernstein coefficients on these regions can be computed economically.