PreprintCC BY 4.0arXiv
On the number of solutions to S-unit equations
Abstract
Let K be a number field and S a finite set of places including the archimedian ones. Let (a1,a2,a3)∈ K* we consider the S-unit equation a1x+a2y=a3 with x,y∈ US. In this paper we prove that for fixed a1,a2 the equation has at most (|S|−1)/2 solutions provided that the height of a3 is sufficiently large. We want to emphasize that our results are effective.
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BibTeX
@misc{ziegler2026number,
title = {{On the number of solutions to $S$-unit equations}},
author = {Volker Ziegler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15487},
eprint = {2609.15487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15487},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the number of solutions to S-unit equations
AU - Volker Ziegler
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15487
UR - https://arxiv.org/abs/2609.15487
AB - Let K be a number field and S a finite set of places including the archimedian ones. Let (a_1,a_2,a_3)∈ K^* we consider the S-unit equation a_1x+a_2y=a_3 with x,y∈ U_S. In this paper we prove that for fixed a_1,a_2 the equation has at most (|S|−1)/2 solutions provided that the height of a_3 is sufficiently large. We want to emphasize that our results are effective.
ER -