The Tropical Non-Properness Set of a Polynomial Map

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