Whitney fold and cusp for algebraic surfaces, and singularities of discriminants

We describe transversal singularity types of the singular locus of the $A$-discriminant for $A\subset\mathbb Z$, and deduce a Whitney type theorem for a coordinate projection of a surface defined by a general polynomial equation with a given Newton polytope $N$: under mild combinaorial conditions on $N$, all multisingularities are stable (folds, cusps, and double folds). We then enumerate the multisingularities in terms of $N$. The results rely on the analysis of degeneracy of relevant Vandermonde/Schur type matrices, which may be of independent interest.