A few remarks on "On certain Vandermonde determinants whose variables separate"

In the recent paper “On certain Vandermonde determinants whose variables separate” [Linear Algebra and its Applications 449 (2014) pp. 17–27], there was established a factorized formula for some bivariate Vandermonde determinants (associated to almost square grids) whose basis functions are formed by Hadamard products of some univariate polynomials. That formula was crucial for proving a conjecture on the Vandermonde determinant associated to Padua-like points. In this note we show that the same formula holds when those polynomials are replaced by arbitrary functions and we extend this formula to general rectangular grids. We also show that the Vandermonde determinants associated to Padua-like points are nonvanishing.