细化多元值集边界
Refining Multivariate Value Set Bounds
摘要：
在有限域上，如果多项式映射的像不是整个域，那么它的基数可以由一个明显更小的值来限制。早期的结果使用多项式的次数限制了值集的基数，但最近的结果使用了所有单项式的幂。在本文中，我们探讨的几何性质的牛顿多面体，并显示它们如何允许更严格的上界的基数的多元值集。然后，我们探索一种方法，它允许更强的上界，无论是否使用多变量度或牛顿多面体的值集。有效地，这提供了Kosters度界的另一种证明，改进的基于牛顿多面体的界，以及Zan和Cao给出的基于度矩阵的结果的改进。
英文摘要：
Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials. In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.
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