Lukasz Nizio

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 1, Pages 76–90

DOI: 10.7546/nntdm.2021.27.1.76-90

**Full paper (PDF, 283 Kb)**

## Details

### Authors and affiliations

Lukasz Nizio

*Faculty of Mathematics and Computer Science, Adam Mickiewicz University
Uniwersytetu Poznanskiego 4, 61-614 Poznan, Poland
*

### Abstract

We construct affine varieties over and imaginary quadratic number fields with a finite number of -lattice points for a fixed , where denotes the ring of algebraic integers of . These varieties arise from equations of the form , where is a rational function, and are polynomials over , and the degree of is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field .

### Keywords

- Diophantine equations
- Algebraic integral points
- Higher dimension affine varieties

### 2010 Mathematics Subject Classification

- 11D45
- 11G35

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## Cite this paper

Nizio, L. (2021). Finiteness of lattice points on varieties *F*(*y*) = *F*(*g*(𝕏)) + *r*(𝕏) over imaginary quadratic fields. *Notes on Number Theory and Discrete Mathematics*, 27(1), 76-90, DOI: 10.7546/nntdm.2021.27.1.76-90.