The Tropical Non-Properness Set of a Polynomial Map

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