多项式映射的热带非完备性集合

Pub Date : 2024-08-08 DOI:10.1007/s00454-024-00684-4
Boulos El Hilany
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引用次数: 0

摘要

我们研究牛顿非退化多项式映射 \(f: {\mathbb {K}}^n \rightarrow {\mathbb {K}}^n\) 的一些离散不变式,这些映射定义在一个代数闭域的普伊塞克斯数列 \({\mathbb {K}}\) 上,并配有一个非三重估值。众所周知,f 不是有限的点的集合 \({\mathcal {S}}(f)\) 在 \({\mathbb {K}}^n\) 中形成了一个代数超曲面。我们证明与 f 对应的热带多项式映射在 f 的热带非良性集上有满足特定组合退化条件的纤维。然后,我们利用这一描述概述了计算该集合的多面体方法,并恢复了复多项式映射非有限性集合的牛顿多面体对偶扇形。这些证明依赖于热带几何中的经典对应和结构结果,并结合了多元结果对 \({mathcal{S}}(f)\)的新描述。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

The Tropical Non-Properness Set of a Polynomial Map

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The Tropical Non-Properness Set of a Polynomial Map

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

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