← 研踪 PaperTrailarXiv:2606.04858 · v2
Integer points close to a transcendental curve: an algorithmic approach
首次提交 2026-06-03 21:25
版本更新 2026-11-03 02:20
雷达观测 2026-11-04 23:41
摘要
In this article, we propose an algorithmic approach to determine the integer points located near a transcendental curve. This approach is closely related to a celebrated work by Bombieri and Pila and to the so-called Coppersmith's method. We establish the underlying theoretical foundations, prove the algorithms, study their complexity and present practical experiments; we also compare our approach with previously existing ones. From a practical point of view, we focus on an instance of our general problem, called the Table Maker's Dilemma, whose solving makes it possible to evaluate a given function with correct rounding. Our experiments show a significant speedup. In particular, our results show that the development of a correctly rounded mathematical library for the binary128 format is now possible at a much smaller cost than with previously existing approaches.
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v2 · 2026-11-03 02:20 · 最新观测版本
Integer points close to a transcendental curve: an algorithmic approach
In this article, we propose an algorithmic approach to determine the integer points located near a transcendental curve. This approach is closely related to a celebrated work by Bombieri and Pila and to the so-called Coppersmith's method. We establish the underlying theoretical foundations, prove the algorithms, study their complexity and present practical experiments; we also compare our approach with previously existing ones. From a practical point of view, we focus on an instance of our general problem, called the Table Maker's Dilemma, whose solving makes it possible to evaluate a given function with correct rounding. Our experiments show a significant speedup. In particular, our results show that the development of a correctly rounded mathematical library for the binary128 format is now possible at a much smaller cost than with previously existing approaches.
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