The zero multiplicity of linear recurrence sequences
type: Item
instance of: scholarly article
title: The zero multiplicity of linear recurrence sequences
zbMATH Open document ID: 0974.11013
DOI: 10.1007/BF02392575
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: A sequence of complex numbers \(U=\{ u_n\}_{n\in{\mathbb{Z}}}\) is said to be a linear recurrence sequence if it satisfies a linear recurrence relation (1) \(u_n=c_1u_{n-1}+\cdots +c_tu_{n-t}\) \((n\in{\mathbb{Z}})\) where \(c_1,\ldots,c_t\) are fixed complex numbers. Let \(t\) be the smallest integer for which \(U\) satisfies a relation of type (1). Then we call \(t\) the order, and \(P_U(z)=z^t-c_1z^{t-1}-\cdots -c_t\) the companion polynomial of \(U\). Write \(P_U(z)=\prod_{i=1}^k (z-\alpha_i)^{a_i}\) where \(\alpha_1,\ldots,\alpha_k\) are the distinct roots of \(P_U\) and \(a_1,\ldots,a_k\) the multiplicities of these roots. Then we can express the terms \(u_n\) of \(U\) as \(u_n=\sum_{i=1}^k f_i(n)\alpha_i^n\) where each \(f_i\) is a polynomial of degree \(a_i-1\). The sequence \(U\) is called non-degenerate if none of the quotients \(\alpha_i/\alpha_j\) \((1\leq i<j\leq k)\) is a root of unity, and it is called simple if \(P_U\) is simple, i.e., if \(a_i=1\) for \(i=1,\ldots,k\). Denote by \(N(U)\) the zero multiplicity of \(U\), that is the number of \(n\in{\mathbb{Z}}\) with (2) \(u_n=0\). Equivalently, \(N(U)\) is the number of solutions of the exponential polynomial equation (3) \(\sum_{i=1}^k f_i(n)\alpha_i^n=0\). Henceforth we assume that \(U\) is non-degenerate. A classical result of Skolem-Mahler-Lech implies that in that case \(N(U)\) is finite. A long outstanding conjecture states that \(N(U)\) is bounded above by a constant depending only on the order \(t\) of \(U\). This conjecture was proved by Schlickewei in the special case that the sequence \(U\) has its terms in the field of rational numbers. More generally, \textit{H. P. Schlickewei} proved that if \(U\) has its terms in an algebraic number field \(K\) of degree \(D\) then \(N(U)\leq c(t,D)\) where \(c(t,D)\) is some effectively computable number depending only on \(t\) and \(D\) [Acta Math. 170, 151-180 (1993; Zbl 0789.11012)]. \textit{F. Beukers} and \textit{H. P. Schlickewei} [Acta Arith. …
reviewed by: Jan-Hendrik Evertse
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 1541207
zbMATH Keywords: linear recurrence sequences
zbMATH Keywords: exponential polynomial equations
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02392575
OpenAlex ID: W2003677899
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description: scientific article; zbMATH DE number 1541207