Given an alternating polynomial $f$ in n variables $x_1, \dots, x_n$, i.e. with $f(\sigma. \textbf x) = \text{sgn}(\sigma)f(\textbf x)$, it is possible to write $f = gV$ with $V$ the Vandermonde polynomial and $g$ a symmetric polynomial.
Is there a nice expression of this symmetric polynomial $g$ in terms of $f$? E.g. simply knowing the coefficients of $f$ when written in a basis of monomials yielding an expression of the coefficients of $g$.
The motivation for this question comes from localization formuli, where these polynomials $g$ appear as equivariant multiplicities.