Singular points of complex algebraic hypersurfaces

We consider a complex hypersurface $V$ given by an algebraic equation in $k$ unknowns, where the set $A\subset {\mathbb Z}^k$ of monomial exponents is fixed, and all the coefficients are variable. In other words, we consider a family of hypersurfaces in $({\mathbb C \setminus 0}) ^ {k}$ parametrized by its coefficients $a =(a_{\alpha})_{\alpha \in A} \in {\mathbb C} ^{A}$. We prove that when $A$ generates the lattice $\mathbb Z^k$ 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 $a$.