The tropical non-properness set of a polynomial map

TL;DR

This paper investigates the tropical non-properness set of polynomial maps over Puiseux series, revealing its structure via tropical geometry and providing methods for its computation and relation to Newton polytopes.

Contribution

It introduces a new description of the non-properness set using multivariate resultants and develops a polyhedral approach for its computation.

Findings

The tropical non-properness set corresponds to fibers with a degeneracy condition.

A polyhedral method for computing the set is outlined.

The set relates to the dual fan of the Newton polytope of non-finite points.

Abstract

We study some discrete invariants of Newton non-degenerate polynomial maps $f : \mathbb{K}^n \to \mathbb{K}^n$ 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.