The singular locus of a tropical hypersurface

A. DICKENSTEIN Y L. F. TABERA

Pucón

Congreso; First joint meeting AMS-SOMACHI; 2010

AMS-SOMACHI

The concept of a singular point of a tropical variety is not well established yet. In the simpler case of hypersurfaces, a possible definition is the following (considered by Markwig, Markwig and Shustin in 2009 in the case of planar curves). Let K be an algebraically closed field of characteristic 0 with a real valuation (with residual field of characteristic 0). We say that a point q in a tropical hypersurface V  in R^d is singular, if there exists a singular algebraic hypersuface of the torus (K^*)^d with tropicalization V , with a singular point of valuation q. We present an equivalent formulation when V is defined by a tropical polynomial with prescribed support A. In principle, this concept, as well as the concept of tropical tangency, can be addressed via the tropicalization of the discriminant variety associated to the family of hypersurfaces with support A (described  in collaboration with Feichtner and Sturmfels, 2007). We give a direct definition of tropical singular point in terms of analogs of Euler derivatives of the tropical polynomial. This allows us to give an algorithm (similar to the algorithm presented by Ochse), to decide whether V is singular and to detect all the singular points. It is not possible to find a simple combinatorial formula to describe all singular points because the situation is not, as one could expect, completely local (cf. the concept of
equivalence). We give a sufficient combinatorial condition for a vertex of a tropical hypersurface to be singular.