A. Esterov

L. Vladimirov

A. Yuran

A Whitney-type cusp and fold theorem is proved for general algebraic surface projections and for sparse univariate polynomial discriminants. 

A classical theorem of Hassler Whitney shows that a generic projection of a smooth surface onto a plane produces only folds, cusps and double folds. Here, we ask how far the result survives for algebraic surfaces. Under mild conditions on the Newton polytope, we prove that the same three singularities exhaust the possibilities for general coordinate projection, and derive formulas to count them from the polytope. 

We describe transversal singularity types of the singular locus of the A-discriminant for A⊂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.

Whitney revisited

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