Key points are not available for this paper at this time.
Abstract We study some discrete invariants of Newton non-degenerate polynomial maps f: {K}ⁿ {K}ⁿ 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 {K}ⁿ K n. The coordinate-wise valuation of {S} (f) ({K}^*) ⁿ S (f) ∩ (K ∗) n is a piecewise-linear object in {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 {S} (f) S (f) in terms of multivariate resultants.
Boulos El Hilany (Thu,) studied this question.