## 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:

  Baker, A.; Masser, D.W., ()  Brownawell, W.D.; Masser, D.W., Vanishing sums in function fields, (), 427-434 · Zbl 0612.10010  Evertse, J.H., On sums of S-units and linear recurrences, Compositio math., 53, 225-244, (1984) · Zbl 0547.10008  Lang, S., ()  Laurent, M., Équations diophantiennes exponentielles, Invent. math., 78, 299-327, (1984) · Zbl 0554.10009  {\scM. Laurent}, Équations exponentielles-polynômes et suites récurrentes linéaires, Astérique, {\bf147-148}, 121-139.  Mason, R.C., Norm form equations, I, J. number theory, 22, 190-207, (1986) · Zbl 0578.10021  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  Schlickewei, H.P.; van der Poorten, A.J., (), Report 82.0041  Shorey, T.N.; Tijdeman, R., ()  van der Poorten, A.J., Additive relations in number fields, (), 259-266 · Zbl 0562.10015
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