Is there any thing in the literature that discusses zeros of Schur functions over $\mathbb{C^n}$? When i say Schur function I mean the one you obtain by dividing the generalized Vandermonde determinant by the principle Vandermonde determinant. Here the generalized Vandermonde determinant is one which has $m_1$$<...<$$m_n$ being a set of increasing integers as exponents of the entries, and all $m_i>0$.