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Integer points close to a transcendental curve: an algorithmic approach

  • Nicolas Brisebarre
  • Guillaume Hanrot
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Abstract

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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Cite

BibTeXRIS
BibTeX
@misc{brisebarre2026integer,
  title = {{Integer points close to a transcendental curve: an algorithmic approach}},
  author = {Nicolas Brisebarre and Guillaume Hanrot},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.04858},
  eprint = {2606.04858},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04858},
}
RIS
TY  - UNPB
TI  - Integer points close to a transcendental curve: an algorithmic approach
AU  - Nicolas Brisebarre
AU  - Guillaume Hanrot
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2606.04858
UR  - https://arxiv.org/abs/2606.04858
AB  - 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.
ER  -