The equation \(ax+by=1\) in characteristic \(p\) (Q1277203)

Let \(K\) be a field and \(G\) a finite rank subgroup of the multiplicative group \(K^* \times K^*\). Consider the equation (*) \(ax+by=1\) with \(a,b\in K^*\) to be solved in \((x,y)\in G\). In case that \(K\) has characteristic \(0\), Schlickewei showed that the number of solutions of (*) is bounded by a quantity depending only on the rank \(r\) of \(G\) and later \textit{F. Beukers} and \textit{H. Schlickewei} obtained the upper bound \(2^{8r+16}\) [Acta Arith. 78, 189-199 (1996; Zbl 0880.11034)]. The proofs of Schlickewei and Beukers and Schlickewei use diophantine approximation techniques which can not be carried over to positive characteristic. Furthermore, equation (*) need not even have finitely many solutions if the characteristic \(p\) of \(K\) is positive. For instance if \(a=b=1\) and \((x,y)\) is a solution of (*) then so is \((x^{p^n},y^{p^n})\) for every positive integer \(n\). The author proves the following result. Suppose that \(K\) has characteristic \(p>0\). Let \(G\) be a finite type subgroup of \(K^*\times K^*\) of rank \(r\). Then (*) has at most \(p^r(p^r+p-2)/(p-1)\) solutions \((x,y)\in G\) unless \((a^n,b^n)\in G\) for some \(n\geq 1\). The author gives an elementary proof. At the end of his paper he discusses some special cases of equation (*) in which the number of solutions can be bounded independently of the characteristic \(p\).