# zbMATH — the first resource for mathematics

On S-unit equations in two unknowns. (English) Zbl 0662.10012
Wide 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.

##### MSC:
 11D57 Multiplicative and norm form equations 11R99 Algebraic number theory: global fields
##### Keywords:
unit equation; at most 2 solutions; S-unit equations
Full Text:
##### References:
  Baker, A.: The theory of linear forms in logarithms. Transcendence theory: advances and applications, pp. 1-27. London: Academic Press 1977  Erdös, P., Stewart, C.L., Tijdeman, R.: Some diophantine equations with many solutions. Compos. Math. to appear · Zbl 0639.10014  Evertse, J.-H.: On equations inS-units and the Thue-Mahler equation. Invent Math.75, 561-584 (1984) · Zbl 0521.10015 · doi:10.1007/BF01388644  Evertse, J.-H.: On sums ofS-units and linear recurrences. Compos. Math.53, 225-244 (1984) · Zbl 0547.10008  Evertse, J.-H., Györy, K.: On the numbers of solutions of weighted unit equations. Compos. Math. (to appear) · Zbl 0644.10015  Evertse, J.-H., Györy, K., Stewart, C.L., Tijdeman, R.:S-unit equations and their applications. New advances in transcendence theory. Cambridge University Press (to appear) · Zbl 0658.10023  Györy, K.: On the number of solutions of linear equations in units of an algebraic number field. Comment. Math. Helv.54, 583-600 (1979) · Zbl 0437.12004 · doi:10.1007/BF02566294  Györy, K.: On the solutions of linear diophantine equations in algebraic integers of bounded norm. Ann. Univ. Budapest Eötvös, Sect. Math.22-23, 225-233 (1979-80)  Lang, S.: Integral points on curves. Publ. Math. Inst. Hautes Etudes Sci.6, 27-43 (1960) · Zbl 0112.13402 · doi:10.1007/BF02698777  Lang, S.: Algebraic number theory. Reading, Mass: Addison-Wesley 1970 · Zbl 0211.38404  Lang, S.: Fundamentals of diophantine geometry. New York: Springer 1983 · Zbl 0528.14013  Laurent, M.: Equations diophantiennes exponentielles. Invent. Math.78, 299-327 (1984) · Zbl 0554.10009 · doi:10.1007/BF01388597  Poorten, A.J., van der: Linear forms in logarithms in thep-adic case. Transcendence theory: advances and applications, pp. 29-57. London: Academic Press 1977  Poorten, A.J., van der, Schlickewei, H.P.: The growth conditions for recurrence sequences. Macquarie Univ. Math. Rep. 82-0041, North. Ryde, Australia, 1982  Shorey, T.N., Tijdeman, R.: Exponential diophantine equations. Cambridge University Press 1986 · Zbl 0606.10011  Siegel, C.L.: Abschätzung von Einheiten. Nachr. Akad. Wiss. Gött., II, Math.-Phys. Kl 71-86, 1969  Zimmert, R.: Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung. Invent. Math.62, 367-380 (1981) · Zbl 0456.12003 · doi:10.1007/BF01394249
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.