Divisibility properties of generalized Vandermonde determinants

Given n ≥ 2 let a denote an increasing n-tuple of non-negative integers ai and let x denote an n-tuple of indeterminates xi. Denote by Va(x) the generalized Vandermonde determinant, the polynomial obtained by computing the determinant of the matrix with (i, j) entry equal to x aj i . Let s be the standard n-tuple of consecutive integers from the interval [0, n−1] and given c ≥ 1 assume that x is an n-tuple of distinct 2-integral odd rational numbers xi such that xi ≡ xj ( mod 2 ). Several years ago one of the authors, investigating some properties of KubotaLeopoldt 2-adic L-functions, asked whether for any n-tuples a and x with c = 1 the identity ord2Va(x) = ord2Vs(x) + ord2Vs(a)− ord2Vs(s) (1.1)