We define a polyhedral version of a stratification for projection maps that applies to any complex or real toric variety and show that it yields similarly desirable properties to the classical map stratification of a proper map. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all the examples we consider, our resulting algorithm outperforms known general purpose methods, e.g., Helmer and Nanda (FoCM, 2022), and Đinh and Jelonek (DCG, 2021), for computing map stratifications.

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.