Equations for the overlaps of a SIC

We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other known systems of equations that determine a Weyl-Heisenberg SIC involve variables and their complex conjugates. We also give a related interesting result about the powers of the projective Fourier transform of the group G = Z d x Z d .