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PreprintCC BY-SA 4.0arXiv

Positive integral points on Mordell-Schinzel surfaces

  • Antoine de Saint Germain
  • Robin Zhang
Open at sourcePDFDOICite

Abstract

We apply the theory of generalised cluster algebras to classify the finiteness of positive integral points on the class of log K3 surfaces defined by xyz = G(x,y) where G has positive coefficients. We furthermore establish results on Zariski density, number of orbits and asymptotic growth of such points. These provide a Diophantine realisation of the finite-affine-indefinite trichotomy of rank-two root systems.

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Cite

BibTeXRIS
BibTeX
@misc{germain2026positive,
  title = {{Positive integral points on Mordell-Schinzel surfaces}},
  author = {Antoine de Saint Germain and Robin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03036},
  eprint = {2611.03036},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03036},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Positive integral points on Mordell-Schinzel surfaces
AU  - Antoine de Saint Germain
AU  - Robin Zhang
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03036
UR  - https://arxiv.org/abs/2611.03036
AB  - We apply the theory of generalised cluster algebras to classify the finiteness of positive integral points on the class of log K3 surfaces defined by xyz = G(x,y) where G has positive coefficients. We furthermore establish results on Zariski density, number of orbits and asymptotic growth of such points. These provide a Diophantine realisation of the finite-affine-indefinite trichotomy of rank-two root systems.
ER  -