The Tropical Non-Properness Set of a Polynomial Map

Abstract

We study some discrete invariants of Newton non-degenerate polynomial maps $$f: {\mathbb {K}}^n \rightarrow {\mathbb {K}}^n$$ f : K n → K n defined over an algebraically closed field of Puiseux series $${\mathbb {K}}$$ K , equipped with a non-trivial valuation. It is known that the set $${\mathcal {S}}(f)$$ S ( f ) of points at which f is not finite forms an algebraic hypersurface in $${\mathbb {K}}^n$$ K n . The coordinate-wise valuation of $${\mathcal {S}}(f)\cap ({\mathbb {K}}^*)^n$$ S ( f ) ∩ ( K ∗ ) n is a piecewise-linear object in $${\mathbb {R}}^n$$ R n , which we call the tropical non-properness set of f. We show that the tropical polynomial map corresponding to f has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of f. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of $${\mathcal {S}}(f)$$ S ( f ) in terms of multivariate resultants.