On positivity of principal minors of bivariate Bezier collocation matrix

TL;DR

This paper investigates the positivity of principal minors of the bivariate Bezier collocation matrix, confirming the conjecture for degrees up to 17 and specific configurations, advancing understanding of polynomial interpolation.

Contribution

It proves the positivity conjecture for the matrix minors for degrees up to 17 and certain point configurations, extending previous theoretical results.

Findings

Confirmed positivity of the matrix minors for degree <=17

Validated conjecture for specific domain point configurations

Contributed to solving the constrained interpolation problem

Abstract

It is well known that the bivariate polynomial interpolation problem at domain points of a triangle is correct. Thus the corresponding interpolation matrix $M$ is nonsingular. L.L. Schumaker stated the conjecture, that the determinant of $M$ is positive. Furthermore, all its principal minors are conjectured to be positive, too. This result would solve the constrained interpolation problem. In this paper, the basic conjecture for the matrix $M$, the conjecture on minors of polynomials for degree <=17 and for some particular configurations of domain points are confirmed.