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Diagonalizable Thue equations: revisited

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Abstract

Let \(r,h\in {\mathbb {N}}\) with \(r\ge 7\) and let \(F(x,y)\in {\mathbb {Z}}[x ,y]\) be a binary form such that

$$\begin{aligned} F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r, \end{aligned}$$

where \(\alpha \), \(\beta \), \(\gamma \), and \(\delta \) are algebraic constants with \(\alpha \delta -\beta \gamma \ne 0\). We establish upper bounds for the number of primitive solutions to the Thue inequality \(0<|F(x, y)| \le h\), improving an earlier result of Siegel and of Akhtari, Saradha, and Sharma.

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Acknowledgements

Saradha would like to thank the Indian National Science Academy for awarding the Senior Scientist fellowship under which this work was done. She also thanks DAE-Center for Excellence in Basic Sciences, Mumbai University for providing facilities to carry out this work. We thank the referee for advising us to point out where the proof fails when \(r=6\).

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Authors and Affiliations

  1. DAE Centre for Excellence in Basic Sciences, University of Mumbai, Kalina campus, Mumbai, 400098, India

    N. Saradha

  2. Department of Mathematics, Birla Institute of Technology and Science, Pilani, 333 031, India

    Divyum Sharma

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  1. N. Saradha
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  2. Divyum Sharma
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Correspondence to Divyum Sharma.

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Divyum acknowledges the support of the DST-SERB SRG Grant SRG/2021/000773 and the OPERA award of BITS Pilani.

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Saradha, N., Sharma, D. Diagonalizable Thue equations: revisited. Ramanujan J 62, 291–306 (2023). https://doi.org/10.1007/s11139-022-00682-1

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