论文标题
Khovanskii的定理和对集合结构的有效结果
Khovanskii's theorem and effective results on sumset structure
论文作者
论文摘要
由于Khovanskii而引起的一个显着定理断言，对于任何有限的子集$ a $ a abelian群体，$ h $ fold的集中$ ha $的基数像多项式一样，对于所有足够大的$ h $。当前，多项式或多项含义均未理解。在本文中，我们为任何$ a \ subset \ mathbb {z}^d $ susevex hull是单纯子的$ a \ subset \ mathbb {z}^d $获得了有效的Khovanskii定理。以前，此类结果仅适用于$ d = 1 $。我们的方法不仅提供了有关$ ha $的基数的信息，而且还提供了其结构，我们证明了两个有效的定理将$ ha $描述为一组：一个回答了Granville和Shakan最近提出的一个问题，另一个是Brion型公式，为所有大型$ H $提供了$ HA $的紧凑描述。作为我们方法的进一步说明，我们在$ | ha | $中得出一个完全明确的公式，只要$ a \ subset \ mathbb {z}^d $由$ d+2 $点组成。
A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. Currently, neither the polynomial nor what sufficiently large means are understood. In this paper we obtain an effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}^d$ whose convex hull is a simplex; previously, such results were only available for $d=1$. Our approach gives information about not just the cardinality of $hA$, but also its structure, and we prove two effective theorems describing $hA$ as a set: one answering a recent question posed by Granville and Shakan, the other a Brion-type formula that provides a compact description of $hA$ for all large $h$. As a further illustration of our approach, we derive a completely explicit formula for $|hA|$ whenever $A \subset \mathbb{Z}^d$ consists of $d+2$ points.
