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Équations exponentielles-polynômes et suites récurrentes linéaires. II. (Exponential polynomial equations and linear recurrence sequences. II). (French) Zbl 0661.10027

Let C be a field of characteristic 0, r a positive integer, L a finite set of subscripts, \(P_{\ell}(X)\in C[X_ 1,...,X_ r]\) for \(\ell \in L\), and \(\chi_{\ell}(X)\) (\(\ell \in L)\) expressions of the form \(b^{X_ 1}_{\ell_ 1}...b^{X_ r}_{\ell_ r}\), with \(b_{\ell 1},...,b_{\ell r}\in C^*\). Partition L into pairwise disjoint subsets \(L_ 1,...,L_ m\). The author considers systems of so- called exponential-polynomial equations \[ (1)\quad \sum_{\ell \in L_ i}P_{\ell}(x)\chi_{\ell}(x)=0\quad in\quad x\in {\mathbb{Z}}^ r \] for \(i=1,...,m\). By a partition \({\mathcal P}\) of \((L_ 1,...,L_ m)\) we mean a collection of pairwise disjoint non-empty sets \(L_{ij}\) \((i=1,...,m\), \(j\in J_ i)\) such that \(L_ i=\cup_{j\in J_ i}L_{ij}\) for \(i=1,...,m\). A solution x of (1) is called \({\mathcal P}\)-compatible if \[ (2)\quad \sum_{\ell \in L_{ij}}P_{\ell}(x)\chi_{\ell}(x)=0\quad for\quad i=1,...,m,\quad j\in J_ i \] and if (2) does not remain valid if \({\mathcal P}\) is replaced by a finer partition. Let \(H_{{\mathcal P}}\) be the subgroup of \({\mathbb{Z}}^ r\) defined by \(\chi_{\ell}(x)=\chi_{\ell '}(x)\) for each pair \(\ell,\ell '\) belonging to the same subset \(L_{ij}\) for \(i=1,...,m\), \(j\in J_ i\). Let \(S_{{\mathcal P}}\) be the set of \({\mathcal P}\)-compatible solutions of (1) such that \(P_{\ell}(x)\neq 0\) for each \(\ell \in L\). For \(x=(x_ 1,...,x_ r)\in {\mathbb{Z}}^ r\) put \(| x| =\max (| x_ 1|,...,| x_ r|)\). The author’s main result is as follows:
Theorem 1. Every \(x\in S_{{\mathcal P}}\) can be expressed as \(x'+x''\), where \(x''\in H_{{\mathcal P}}\) and \(| x'| \leq C\) if all polynomials \(P_{\ell}\) are constant and \(| x'| \leq C \log | x|\) otherwise, where C is a constant independent of x.
In an earlier paper, the author proved this result for the case that C is an algebraic number field [Invent. Math. 78, 299-327 (1984; Zbl 0554.10009)]. The author obtains the general result by applying a result of W. D. Brownawell and D. W. Masser on S-unit equations over function fields [Math. Proc. Camb. Philos. Soc. 100, 427-434 (1986; Zbl 0612.10010)] and some specialization arguments. Using his Theorem 1, the author proves the following result on linear recurrence sequences \(u_ n=\sum^{k}_{i=1}f_ i(n)\alpha^ n_ i\) (n\(\in {\mathbb{Z}})\), where \(k\geq 2\), \(f_ i(X)\in C[X]\) for \(i=1,...,n\), \(\alpha_ 1,..,\alpha_ n\in C^*\) and \(\alpha_ i/\alpha_ j\) is not a root of unity for \(1\leq i<j\leq k\). For each \(\lambda \in C^*\), let \(S_{\lambda}\) be the set of pairs \((m,n)\in {\mathbb{Z}}^ 2\) such that \(u_ m=\lambda u_ n\). Put \(D=\{(m,n)\in {\mathbb{Z}}^ 2:\) \(m=n\}\), \(H=\{(m,n)\in {\mathbb{Z}}^ 2:\) \(m=-n\}.\)
Theorem 2. \(S_ 1\) is the union of a finite set, D, and possibly a set of the form \((a,b)+rH\) for fixed integers a,b,r; if \(\lambda\) is a root of unity \(\neq 1\) then \(S_{\lambda}\) is the union of a finite set and possibly a set of the form \((a,b)+rH\) with fixed a,b,r\(\in {\mathbb{Z}}\); if \(\lambda\) is not a root of unity then \(S_{\lambda}\) is finite.
Reviewer: J.-H.Evertse

MSC:

11D61 Exponential Diophantine equations
11B37 Recurrences
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