Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

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
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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 \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r 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 \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

Keywords

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

2010 Mathematics Subject Classification

  • 11D45
  • 11G35

References

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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.

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