On the Vandermonde Determinant of Padua-like Points

Recently [1] 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(n))2). One may observe that these points have the structure of the union of two (tensor product) 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 functions. We offer 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.