论文标题
多项式图的热带非构图集
The tropical non-properness set of a polynomial map
论文作者
论文摘要
我们研究了牛顿非分类多项式映射$ f的一些离散不变的，\ mathbb {k}^n \ to \ mathbb {k}^n $定义在puiseux系列$ \ mathbb {k} $的代数封闭字段上，配备了非trivivial估算。众所周知，$ f $不是有限的点$ \ mathcal {s}（f）$，在$ \ mathbb {k}^n $中形成代数hypersurface。 $ \ mathcal {s}（f）\ cap（\ mathbb {k}^*）^n $的坐标估值是$ \ mathbb {r}^n $中的分段线性对象，我们称之为热带非培训集$ f $。我们表明，与$ f $相对应的热带多项式地图具有满足特定组合的堕落条件的纤维，正好超过了$ f $的热带非传播集中的点。然后，我们使用此描述来概述一种多面体方法来计算此集合，并将风扇双重恢复到该集合的Newton Polytope中，该集合中的复杂多项式映射不是有限的。这些证明依赖于热带几何形状的经典对应关系和结构性结果，并在多变量结果方面结合了$ \ Mathcal {s}（f）$的新描述。
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.
