Singular Points of Complex Algebraic Hypersurfaces

We consider a complex hypersurface V given by an algebraic equation in k unknowns, where the set A ⊂ Zk of monomial exponents is fixed, and all the coefficients are variable. In other words, we consider a family of hypersurfaces in (C \ 0)k parametrized by its coefficients a = (aα)α∈A ∈ CA. We prove that when A generates the lattice Zk as a group, then over the set of regular points a in the A-discriminantal set, the singular points of V admit a rational expression in a.