On positivity of principal minors of bivariate Bezier collocation matrix
For researchers in multivariate spline interpolation, this work provides partial confirmation of a long-standing conjecture, but the result is incremental as it only covers limited cases.
The paper confirms Schumaker's conjecture that the bivariate Bezier collocation matrix has positive determinant and positive principal minors for polynomial degrees up to 17 and certain domain point configurations, which would enable constrained interpolation.
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.