Counting solutions to trinomial Thue equations: a different approach

Thomas, Emery

doi:10.1090/S0002-9947-00-02437-5

Counting solutions to trinomial Thue equations: a different approach
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Trans. Amer. Math. Soc. 352 (2000), 3595-3622 Request permission

Abstract:

We consider the problem of counting solutions to a trinomial Thue equation — that is, an equation \begin{equation*} |F(x,y)| = 1,\tag {$*$} \end{equation*} where $F$ is an irreducible form in $Z[x,y]$ with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the “Thue-Siegel principle" and its relation to $(*)$. In this paper we give specific numerical bounds for the number of solutions to $(*)$ by a somewhat different approach, the difference lying in the initial step — solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.

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