Lattices in function fields and applications

IF 0.8 3区 数学 Q2 MATHEMATICS Mathematika Pub Date : 2025-01-31 DOI:10.1112/mtk.70010
Christian Bagshaw, Bryce Kerr
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引用次数: 0

Abstract

In recent decades, the use of ideas from Minkowski's Geometry of Numbers has gained recognition as a helpful tool in bounding the number of solutions to modular congruences with variables from short intervals. In 1941, Mahler introduced an analogue to the Geometry of Numbers in function fields over finite fields. Here, we build on Mahler's ideas and develop results useful for bounding the sizes of intersections of lattices and convex bodies in , which are more precise than what is known over . These results are then applied to various problems regarding bounding the number of solutions to congruences in , such as the number of points on polynomial curves in low-dimensional subspaces of finite fields. Our results improve on a number of previous bounds due to Bagshaw, Cilleruelo, Shparlinski and Zumalacárregui. We also present previous techniques developed by various authors for estimating certain energy/point counts in a unified manner.

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来源期刊
Mathematika
Mathematika MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.40
自引率
0.00%
发文量
60
审稿时长
>12 weeks
期刊介绍: Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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