On positivity of principal minors of bivariate Bézier collocation matrix

Abstract

It is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct. Thus the corresponding interpolation matrix M is nonsingular. Schumaker stated the conjecture that all principal submatrices of M are nonsingular too. Furthermore, all of the corresponding determinants (the principal minors) are conjectured to be positive. This result would solve the constrained interpolation problem. In this paper, the conjecture on minors for polynomial degree ⩽17 and conjecture for some particular configurations of domain points are confirmed.