Linear equations over multiplicative groups, recurrences, and mixing. I (Q2888913)

The paper takes its origin from two previous results of the authors.NEWLINENEWLINEIn [Isr. J. Math. 142, 189--204 (2004; Zbl 1055.37009)], \textit{D. W. Masser} considered linear equations over multiplicative groups in positive characteristic, and introduced \(n-1\) independently operating Frobenius maps, where \(n\) is the number of variables.NEWLINENEWLINEIn [Invent. Math. 168, No. 1, 175--224 (2007; Zbl 1205.11030)], \textit{H. Derksen} studied recurrences in positive characteristic, and introduced integer sequences involving combinations of \(d-2\) powers of the characteristic; here, \(d\) is the order of the recurrence.NEWLINENEWLINEThe authors observe that these two new concepts are identical in pratice, and use this observation to improve on the previous result of Masser, by proving some theorems more closely related to what is known for zero characteristic. If \(K\) is a field of positive characteristic, \(G\) is a finitely generated subgroup of \(K^*\) and \(V\) is a linear variety in \(K^n\), they show how to compute \(V\cap G^n\) effectively, with completely explicit estimates. As a byproduct, they obtain the effective solution of the \(S\)-unit equation in \(n\) variables.