Representation of unity by binary forms

Akhtari, Shabnam

doi:10.1090/S0002-9947-2011-05507-8

Representation of unity by binary forms
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Trans. Amer. Math. Soc. 364 (2012), 2129-2155 Request permission

Abstract:

In this paper, it is shown that if $F(x , y)$ is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of $F$ is large enough, the equation $F(x , y) = \pm 1$ has at most $11n-2$ solutions in integers $x$ and $y$. We will also establish some sharper bounds when more restrictions are assumed. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential role in studying the geometry of our Diophantine equations.

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