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Counting solutions to trinomial Thue equations: a different approach. (English) Zbl 0995.11025

Let \(F(x,y)\) be an irreducible form in \(\mathbb{Z}[x,y]\) with degree \(n\geq 3\), having exactly three non-zero coefficients. The author studies the number \(N_F\) of “regular solutions” to the trinomial Thue equation \(F(x,y)= \pm 1\) (a solution \((x,y)\) is “regular” if \(x\neq 0\), \(y>0\) and \(y\neq|x|\)). By a result of E. Bombieri and M. W. Schmidt on Thue (not necessarily trinomial) equations [Invent. Math. 88, 69-81 (1987; Zbl 0614.10018)], this number is at most constant times \(n\) and the constant can be taken to be 430 if \(n\) is sufficiently large. From the specialized to trinomial Thue equations work of J. Mueller and M. W. Schmidt [J. Reine Angew. Math. 379, 76-99 (1987; Zbl 0613.10019)], it follows that \(N_F\) is bounded by an effective but not explicit universal constant (independent from \(F\) and \(n\)). An explicit general result on Thue-Mahler equations over algebraic number fields, due to J.-H. Evertse [J. Reine Angew. Math. 482, 121-149 (1997; Zbl 0861.11021)], implies the explicit, but very large and dependent on \(n\), upper bound \(N_F\leq 5\cdot 10^6 n\).
The main result of the paper under review implies that, if \(n\geq 5\), then \(N_F\leq 81\) and, on specializing the range and parity of \(n\), the author obtains far better bounds; for example, if \(17\leq n\leq 37\) and \(n\) is odd, then \(N_F\leq 18\). The various specializations for \(n\) are explicitly stated in the paper.
As in the above mentioned papers of Bombieri, Mueller and Schmidt, the so-called “gap principle” and “Thue-Siegel principle” are basic tools. The approach of the present paper mainly differs from that of Bombieri-Mueller-Schmidt in the initial step: To each nontrivial solution \((x,y)= (p,q)\) these authors associate the closest to \(p/q\) root \(\omega\) (real or complex) of \(f(x):= F(x,1)\) and examine how well \(p/q\) approximates \(\omega\), while this author makes the following (we quote from his introduction): “[…] we associate to \((p,q)\) either a real root of \(f(x)\) or a real root of \(f'(x)\) […]. We call the set of all these roots exceptional set of \(F\). If \((p,q)\) is associated with an exceptional point \(\tau\), we again have the question: how well does \(p/q\) approximate the (real) number \(\tau\)? The method used here to solve the approximation problem is quite different […]. We work with polynomials with real coefficients, and we regard a given Thue trinomial as a member of a 1-parameter family of real trinomials. In this family we then select one or two trinomials called maximal. These trinomials have two key properties: first, the approximation problem is readily solved for these trinomials (and the solution is “almost” sharp); second, the solution for these trinomials is “maximal” for all trinomials in the family. In this way we solve the approximation problem for an arbitrary Thue trinomial.”
The author claims that his method can be extended to Thue tetranomials (i.e. those \(F\) with four nonzero coefficients) in a similar way.

MSC:

11D59 Thue-Mahler equations
11D45 Counting solutions of Diophantine equations
11J68 Approximation to algebraic numbers
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References:

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