arXiv 1603.033402016-03math.NT

Thue's inequalities and the hypergeometric method

Shabnam Akhtari, N. Saradha, Divyum Sharma

备注：: 45 pages

英文摘要：

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape $0<|F(x, y)| \leq h$, where $F(x , y) =(αx + βy)^r -(γx + δy)^r \in \mathbb{Z}[x ,y]$, $α$, $β$, $γ$ and $δ$ are algebraic constants with $αδ-βγ\neq 0$, and $r \geq 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue's inequaities $|ax^r - by^r| \leq c$. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.

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