Intersections in subvarieties of ${\mathbb {G}}_{\mathrm {m}}^l$ and applications to lacunary polynomials
Intersections in subvarieties of ${\mathbb {G}}_{\mathrm {m}}^l$ and applications to lacunary polynomials
复制标题
${mathbb {G}}_{mathrm {m}}^l$ 子类型的交集及其在缺陷多项式中的应用
DOI：
10.1090/tran/8470
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发表时间：
2022
影响因子：
1.3
通讯作者：
Zannier, Umberto
中科院分区：
文献类型：
--
作者：
Corvaja, Pietro;Levin, Aaron;Zannier, Umberto
通过函数域上的单位点来解释上下文，我们研究了给定的子变量与1参数子集的陪集的交集。在采用第二作者最近提出的一种方法的函数域版本的基础上，将第一作者和第三作者以前的工作推广到任意维，我们证明了当交集的数目比预期的多时，可以对相关的子环进行分类。作为结果，我们得到了亚拓里陪集的一个分类，使得有许多与的多重交集。这也为Erd、ő、S和Rényi关于缺项多项式的猜想提供了一个新的证明。最后，我们展示了这些方法如何在不可能相交的领域产生结果，并在最后一节中，用截断计数函数重新解释了Vojta猜想的一些结果。参考文献
We investigate intersections of a given subvarietyofwith cosets of 1-parameter subtori, on interpreting the context in terms of-unit points over function fields. On adopting a function field version of a method introduced recently by the second author, extending to arbitrary dimensions previous work of the first and third authors, we prove that when the number of intersections is substantially higher than expected, one can classify the relevant subtori. As a consequence, we obtain a classification of the cosets of subtori such that there are many multiple intersections with. This also allows a new proof of a conjecture of Erdős and Rényi on lacunary polynomials. We finally show how the methods yield results in the realm of Unlikely Intersections in, and in the last section, reinterpret some of the results in terms of Vojta’s conjecture with truncated counting functions. References