On a conjecture concerning interpolation by bivariate Bernstein polynomials

Abstract

In this paper we discuss a conjecture of Schumaker that the principal submatrices of collocation matrices of bivariate Bernstein polynomials over triangular grids have positive determinant. It is easy to show that the conjecture holds for the 2 × 2 submatrices. In this paper we show that it also holds for the 3 × 3 submatrices, working with the equivalent ‘monomial form’ of the conjecture. This result generalizes to a class of 3 × 3 matrices which will be described.