##
**Linear equations in variables which lie in a multiplicative group.**
*(English)*
Zbl 1026.11038

Let \(K\) be a field of characteristic 0 and let \(n\) be a natural number. Let \(\Gamma\) be a subgroup of the multiplicative group \((K^*)^m\) of finite rank \(r\). Given \(a_1,\dots,a_n\in K^*\), let \(A(a_1,\dots,a_n,\Gamma)\) be the number of solutions \(x = (x_1,\dots,x_n)\in \Gamma\) of the equation \(a_1x_1 + \dots + a_nx_n = 1\) such that no proper subsum of \(a_1x_1 + \dots + a_nx_n\) vanishes. The main result of the paper is
\[
A(a_1,\dots,a_n,\Gamma) \leq A(n,r) = \exp((6n)^{3n}(r+1)).
\]
As remarked by the authors, the essential feature of their inequality is uniformity, since \(A(n,r)\) is independent of \(K\) and of the coefficients \(a_1,\dots,a_n\). Moreover, although the upper bound \(A(n,r)\) is probably far from best possible, a number of examples show that a certain dependence on \(n\) and \(r\) is needed.

In particular cases, even better upper bounds for \(A(a_1,\dots,a_n,\Gamma)\) had already been proved: for \(n=2\) one has the estimate \( A(a_1,a_2,\Gamma) \leq 2^{9(r+1)}\) [F. Beukers and H. P. Schlickewei, Acta Arith. 78, 189-199 (1996; Zbl 0880.11034)] and for \(r=0\) and arbitrary \(n\) one has the estimate \(A(a_1,\dots,a_n,\Gamma) \leq (n+1)^{3(n+1)^2}\) [J.-H. Evertse, Acta Arith. 89, 45-51 (1999; Zbl 0974.11012)]. However, for all other cases the result is new.

Also, for \(K\) a number field, \(\Gamma\) may be viewed as a subgroup of the group of \(S\)-units of \(K\) (where \(S\) is a finite set of places of \(K\) including the Archimedean places), and the finiteness of \(A(a_1,\dots,a_n,\Gamma)\) had already been established (the first result is due to Mahler for \(n=2\)). An explicit upper bound was found by Schlickewei and Schmidt, but this upper bound depended not only on \(n\) and \(r\) but also on the degree \(d=[K:{\mathbb Q}]\) of the number field \(K\). The result of the paper finally solves the problem of finding a bound independent of \(d\).

In a second theorem, the authors apply their main result to linear recurrence sequences of order \(n\). It is known after Skolem-Mahler-Lech that, if \(u_m\) is a linear recurrence sequence, the solutions \(k\) of \(u_k=0\) are contained in a finite set of arithmetical progressions plus a finite set. The bound for the number of arithmetical progressions plus the number of points proved in the paper is \(\exp((6n)^{3n})\), hence only doubly exponential in \(n\), improving on an estimate of Schmidt, which is triply exponential.

The proofs use mainly two tools: the quantitative version of the Subspace Theorem, proved by Evertse and Schlickewei, to deal with large solutions and a method of Schmidt, involving lower bounds for the height of points in algebraic varieties, to deal with small solutions.

In particular cases, even better upper bounds for \(A(a_1,\dots,a_n,\Gamma)\) had already been proved: for \(n=2\) one has the estimate \( A(a_1,a_2,\Gamma) \leq 2^{9(r+1)}\) [F. Beukers and H. P. Schlickewei, Acta Arith. 78, 189-199 (1996; Zbl 0880.11034)] and for \(r=0\) and arbitrary \(n\) one has the estimate \(A(a_1,\dots,a_n,\Gamma) \leq (n+1)^{3(n+1)^2}\) [J.-H. Evertse, Acta Arith. 89, 45-51 (1999; Zbl 0974.11012)]. However, for all other cases the result is new.

Also, for \(K\) a number field, \(\Gamma\) may be viewed as a subgroup of the group of \(S\)-units of \(K\) (where \(S\) is a finite set of places of \(K\) including the Archimedean places), and the finiteness of \(A(a_1,\dots,a_n,\Gamma)\) had already been established (the first result is due to Mahler for \(n=2\)). An explicit upper bound was found by Schlickewei and Schmidt, but this upper bound depended not only on \(n\) and \(r\) but also on the degree \(d=[K:{\mathbb Q}]\) of the number field \(K\). The result of the paper finally solves the problem of finding a bound independent of \(d\).

In a second theorem, the authors apply their main result to linear recurrence sequences of order \(n\). It is known after Skolem-Mahler-Lech that, if \(u_m\) is a linear recurrence sequence, the solutions \(k\) of \(u_k=0\) are contained in a finite set of arithmetical progressions plus a finite set. The bound for the number of arithmetical progressions plus the number of points proved in the paper is \(\exp((6n)^{3n})\), hence only doubly exponential in \(n\), improving on an estimate of Schmidt, which is triply exponential.

The proofs use mainly two tools: the quantitative version of the Subspace Theorem, proved by Evertse and Schlickewei, to deal with large solutions and a method of Schmidt, involving lower bounds for the height of points in algebraic varieties, to deal with small solutions.

Reviewer: Roberto Dvornicich (Pisa)