One-parameter families of unit equations (Q862070)

Let \(k\) be an algebraic number field, \(S\) a finite set of places of \(k\) containing the archimedean places, and denote by \(O_{k,S}\), \(O_{k,S}^*\) the ring of \(S\)-integers, group of \(S\)-units in \(k\), respectively. The author considers parametrized classes of unit equations \[ f(t)u+g(t)v=h(t)\quad\text{in \(u,v\in O_{k,S}^*\), \(t\in k\),} \tag{1} \] where \(f,g,h\) are polynomials in \(k[t]\). The author's main result is, that (1) has only finitely many solutions in triples \((u,v,t)\) if \(f,g\) are non-constant, \(\deg h =\deg f+\deg g\) and \(\deg h>2\). Consequently, in this case there are only finitely many \(t\in k\) such that (1) is solvable in \(u,v\in O_{k,S}^*\), and so (1) gives rise to an infinite, parametrized class of \(S\)-unit equations in two variables that are unsolvable. In the case that \(f,g\) are linear and \(h\) is quadratic, the author shows that all but finitely many solutions of (1) are contained in a finite union of certain explicitly given parametrized families. The key idea of the author is to map the solutions of (1) to the set of \(S\)-integral points of \(\mathbb P^1\times\mathbb P^1\setminus Z\), where \(Z\) is a union of four explicitly given curves in \(\mathbb P^1\times\mathbb P^1\). It follows from general work of the author from [Generalizations of Siegel's and Picard's theorems, Ann. Math. (2), to appear] that this set of \(S\)-integral points is the union of a finite set and a finite number of curves which are independent of \(k,S\). A further analysis of these curves leads to the author's results on (1). We mention that the author's result on \(\mathbb P^1\times\mathbb P^1\setminus Z\) is in a similar vain as work of \textit{P. Corvaja} and \textit{U. Zannier} on integral points on surfaces [Ann. Math. (2) 160, 705--726 (2004; Zbl 1146.11035)]. In particular, it is a consequence of the subspace theorem.