## 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.

### MSC:

 11D45 Counting solutions of Diophantine equations 11D04 Linear Diophantine equations

### Citations:

Zbl 0880.11034; Zbl 0974.11012
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