On the Vandermonde Determinant of Padua-like Points (an open problem)

Recently we gave a simple, geometric and explicit construction of bivariate interpolation at points in a square (the so-called Padua points), and showed that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O((log^2(n)). One may observe that these points have the structure of the union of two (tensor prod- uct) grids, one square and the other rectangular. In this article we give a conjectured formula (in the even degree case) for the Vandermonde determinant of any set of points with exactly this structure. Surprisingly, it factors into the product of two univariate func- tions. We oer a partial proof that depends on a certain technical lemma (Lemma 1 below) which seems to be true but up till now a correct proof has been elusive.