The Tropical Non-Properness Set of a Polynomial Map

Abstract

We study some discrete invariants of Newton non-degenerate polynomial maps \(f: {\mathbb {K}}^n \rightarrow {\mathbb {K}}^n\) defined over an algebraically closed field of Puiseux series \({\mathbb {K}}\), equipped with a non-trivial valuation. It is known that the set \({\mathcal {S}}(f)\) of points at which f is not finite forms an algebraic hypersurface in \({\mathbb {K}}^n\). The coordinate-wise valuation of \({\mathcal {S}}(f)\cap ({\mathbb {K}}^*)^n\) is a piecewise-linear object in \({\mathbb {R}}^n\), 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 \({\mathcal {S}}(f)\) in terms of multivariate resultants.