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Saradha, Shabnam Akhtari","submitted_at":"2016-03-10T17:27:53Z","abstract_excerpt":"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 $F(x , y) =(\\alpha x + \\beta y)^r -(\\gamma x + \\delta y)^r \\in \\mathbb{Z}[x ,y]$, $\\alpha$, $\\beta$, $\\gamma$ and $\\delta$ are algebraic constants with $\\alpha\\delta-\\beta\\gamma \\neq 0$, and $r \\geq 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue's inequaities $|ax^r - by^r| \\leq c$. 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