Linear equations over multiplicative groups in positive characteristic
\(S\)-unit equations and related equations over number fields or over function fields of characteristic \(0\), as well as generalizations in the vain of the Mordell-Lang conjecture for semi-abelian varieties, can be treated by techniques from Diophantine approximation, but these techniques are not applicable for fields of positive characteristic. In the past few years, various new techniques have been developed to treat Diophantine equations in positive characteristic, based on abc-type inequalities for function fields of positive characteristic, differentiation of algebraic functions, model theory and finite automata. Building further on earlier individual work of both authors, \textit{H. Derksen} and \textit{D. Masser} [Proc. Lond. Math. Soc. (3) 104, No. 5, 1045--1083 (2012; Zbl 1269.11062)] proved a general result concerning the structure of sets \(V\cap G^n\), where \(V\) is a linear subvariety of \(K^n\) for some field \(K\) of characteristic \(p>0\), \(G\) is a finitely generated subgroup of \(K^*\) and \(G^n\subset (K^*)^n\) is the \(n\)-fold direct product of \(G\). Their argument is elementary, and makes essential use of differentiation of algebraic functions. In the paper under review, the author gives more detailed results in some special cases.NEWLINENEWLINETo describe the above mentioned results, we introduce some further notation. For any power \(q\) of \(p\), we denote by \(\varphi_q\) the Frobenius map \((x_1,\dots , x_n)\mapsto (x_1^q,\dots ,x_n^q)\) on \(K^n\). A \(G\)-automorphism of \(K^n\) is a map \((x_1,\dots ,x_n)\mapsto (g_1x_1,\dots ,g_nx_n)\), where \(g_1,\dots ,g_n\in G\). For any tuple of \(G\)-automorphisms \(\psi_1,\dots ,\psi_h\) we define the operator NEWLINE\[NEWLINE [\psi_1,\dots ,\psi_h]_q :=\bigcup_{e_1=0}^{\infty}\cdots\bigcup_{e_h=0}^{\infty} (\psi_1^{-1}\varphi_q^{e_1}\psi_1)\cdots (\psi_h^{-1}\varphi_q^{e_h}\psi_h) NEWLINE\]NEWLINE which maps each point of \(K^n\) to an infinite subset of \(K^n\); here the operator is just the identity if \(h=0\). By a \(q\)-family we mean a set of the shape \([\psi_1,\dots ,\psi_h]_q(T\cap G^n)\), where \(\psi_1,\dots , \psi_h\) are any \(h\geq 0\) \(G\)-automorphisms of \(K^n\), and \(T\) is a coset of an algebraic subgroup of \({\mathbb G}_m^n\), contained in \(V\). Then the theorem of Derksen and Masser reads as follows: Let \(n\geq 2\) and assume that \(\root K\of G:=\{ \tau\in K^*:\, \exists m\in{\mathbb Z}_{>0}\;\text{with } \tau^m\in G\}\) is finitely generated. Then there is a power \(q\) of \(p\) such that \(V\cap G^n\) is a finite union of \(q\)-families, which can all be determined effectively.NEWLINENEWLINEThe families in the result of Derksen and Masser can be determined effectively in principle, given the prime \(p\) and suitable representations for \(K,V,G\), but their arguments do not provide a practical way to determine them. In the paper under review, combining the ideas of Derksen and Masser with careful case-by-case analyses, the author gives an explicit description of all families in the following cases: \(K={\mathbb F}_p[t]\), \(G=\langle t,1-t\rangle\) the multiplicative group generated by \(t\) and \(1-t\), \(n=2\), \(V\) the linear variety defined by \(x_1+x_2=1\); and \(K={\mathbb F}_p[t]\), \(G=\langle t,1-t\rangle\), \(n=3\) and \(V\) the linear variety defined by \(x_1+x_2-x_3=1\). As it turns out, the families can be described in a manner independent of \(p\) for \(p\geq 5\), while for \(p=2,3\) there are some additional families.
scientific article; zbMATH DE number 6033771