Solving Thue equations of high degree
The purpose of this remarkable paper is clearly indicated by its title. The authors' method is based on the general method for solving in practice Thue equations, as described in the paper, rather widely known to people dealing with diophantine equations, by \textit{N. Tzanakis} and \textit{B. M. M. de Weger} [J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)]. The present authors, however, proceed much further and succeed in solving Thue equations that the reviewer, at least, did not ever hope to solve by the method of the above-mentioned paper. The reason is the following: Let \(r\) be the number of fundamental units of the number field, which is involved in the solution of the Thue equation. Then, certain linear forms (actually, very few in practice) in \(r+1\) logarithms of (known) algebraic numbers with \(r\) (unknown) integer coefficients enter into the game. The upper bound for the unknown integer coefficients is obtained by a Baker-type result and is so large that one can never hope to check by enumeration all possibilities. Therefore, techniques for the reduction of this huge upper bound had to be invented. Two of them are widely known: one is based on the so-called Baker-Davenport Lemma and the other, based on the LLL basis reduction algorithm; the second being more widely applied, especially when \(r\geq 3\). The reviewer and B. M. M. de Weger, in their aforementioned paper, adopt the second reduction technique. They work in a certain lattice of dimension \(r\) and apply the LLL-algorithm in order to find a reduced basis. This is the crucial step for reducing the upper bound. When, however, \(r\) is very large, like in the examples solved in the paper under review \((r=9,32)\), there is no hope for the LLL-algorithm to work, given also that the coordinates of the lattice vectors are huge integers. In the paper under review, rather simple and natural observations lead to the impressive conclusion that, roughly speaking, instead of considering linear forms in \(r+1\) logarithms, one can work with \(r\) linear forms in two (known) logarithms with two (unknown) integer coefficients. This clearly makes reduction a feasible task. Specific examples of degree 19 and 33 are solved. The degree 19 equations are \[ 2x^{19}+ y^{19}=\pm1,\pm2; \] here, \(r=9\). The degree 33 equations are \[ F_{67}(x,y)= \pm1,\pm67,\quad\text{where }F_{67}= \prod^{33}_{k=1}\Biggl(y-x\cdot\cos {2\pi k\over 67}\Biggr) \] is a polynomial with integer coefficients, the largest of which is 1144066. In this example \(r=32\).
scientific article; zbMATH DE number 955265