## 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

### Citations:

Zbl 0554.10009; Zbl 0612.10010
Full Text:

### References:

 [1] Baker, A.; Masser, D.W., () [2] Brownawell, W.D.; Masser, D.W., Vanishing sums in function fields, (), 427-434 · Zbl 0612.10010 [3] Evertse, J.H., On sums of S-units and linear recurrences, Compositio math., 53, 225-244, (1984) · Zbl 0547.10008 [4] Lang, S., () [5] Laurent, M., Équations diophantiennes exponentielles, Invent. math., 78, 299-327, (1984) · Zbl 0554.10009 [6] {\scM. Laurent}, Équations exponentielles-polynômes et suites récurrentes linéaires, Astérique, {\bf147-148}, 121-139. [7] Mason, R.C., Norm form equations, I, J. number theory, 22, 190-207, (1986) · Zbl 0578.10021 [8] Mignotte, M.; Shorey, T.N.; Tijdeman, R., The distance between terms of an algebraic recurrence sequence, J. reine angew. math., 349, 63-76, (1984) · Zbl 0521.10011 [9] Schlickewei, H.P.; van der Poorten, A.J., (), Report 82.0041 [10] Shorey, T.N.; Tijdeman, R., () [11] van der Poorten, A.J., Additive relations in number fields, (), 259-266 · Zbl 0562.10015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.