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arXiv · 2505.16465

Explicit and Mixed Estimates for Thue inequalities with few coefficients

N. Saradha·Divyum Sharma
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Abstract

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were written giving an upper bound for the number of solutions of the above inequality as $\ll c(r,s,h)$ where $c(r,s,h)$ is an explicit function of $r,s$ and $h.$ Invariably, the absolute constant involved in $\ll$ has been left undetermined. In this paper, following Bombieri, Schmidt and Mueller, we give three different upper bounds which are explicit in every aspect.

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N. Saradha, Divyum Sharma. 2025-05-22. Explicit and Mixed Estimates for Thue inequalities with few coefficients. https://arxiv.org/abs/2505.16465

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