On S-unit equations in two unknowns.

*(English)*Zbl 0662.10012Wide classes of diophantine equations reduce ultimately to the solution of equations of the shape \(\alpha_ 1x+\alpha_ 2y=\alpha_ 3\) with the \(\alpha_ i\) in \({\mathbb{K}}^{\times}\), \({\mathbb{K}}\) some given number field and x, y generalised units (in the present terminology: S-units) of \({\mathbb{K}}\). There is an obvious equivalence relation on the triples \((\alpha_ 1,\alpha_ 2,\alpha_ 3)\) so that the cited equation retains the same number of solutions. The first theorem of this paper is the remarkable result that, with the exception of finitely many equivalence classes, the unit equation has at most 2 solutions. This result is an ingenious consequence of the work on general S-unit equations of the reviewer and Schlickewei and independently of Evertse.

The idea is that if the equation has as many as three solutions, then, by elementary linear algebra, one has an S-unit equation in the solutions and that entails severe restriction on the \(\alpha_ i\). Thus the main result is ineffective but a further result, with the exclusions in principle determinable, yields a general bound of \(s+1\), where s denotes the cardinality of the nonarchimedean places defining S. Those equivalence classes with many solutions have a representative for which all solutions have height less than a cited explicitly computable bound. In contrast, it is known that there cannot be a general upper bound polynomial in s for the number of solutions of the S-unit equation.

The reader should also study the survey by the present authors [New advances in transcendence theory, Proc. Sympos., Durham/UK 1986, 110-174 (1988; Zbl 0658.10023)] in which the present ideas were first noticed.

The idea is that if the equation has as many as three solutions, then, by elementary linear algebra, one has an S-unit equation in the solutions and that entails severe restriction on the \(\alpha_ i\). Thus the main result is ineffective but a further result, with the exclusions in principle determinable, yields a general bound of \(s+1\), where s denotes the cardinality of the nonarchimedean places defining S. Those equivalence classes with many solutions have a representative for which all solutions have height less than a cited explicitly computable bound. In contrast, it is known that there cannot be a general upper bound polynomial in s for the number of solutions of the S-unit equation.

The reader should also study the survey by the present authors [New advances in transcendence theory, Proc. Sympos., Durham/UK 1986, 110-174 (1988; Zbl 0658.10023)] in which the present ideas were first noticed.

Reviewer: A.J.van der Poorten

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