Multiplicities of recurrence sequences (Q1373001)

Let \(\nu_0,\ldots ,\nu_{n-1}\) with \(n\geq 2\) be algebraic numbers such that \(\nu_0\not=0\), and let \(U=\{ u_m\}_{m\in{\mathbb{Z}}}\) be a non-identically zero sequence of algebraic numbers satisfying the recurrence relation \(u_{m+n}=\nu_{n-1}u_{m+n-1}+\cdots +\nu_0u_m\) for \(m\in{\mathbb{Z}}\). Assume in what follows that the sequence \(U\) is non-degenerate, that is, if the companion polynomial \(G(z):=z^n-\nu_{n-1}z^{n-1}-\cdots -\nu_0\) of the recurrence relation factorises as \(\prod_{i=1}^q (z-\alpha_i)^{e_i}\), where \(\alpha_1,\ldots ,\alpha_q\) are distinct algebraic numbers and \(e_1,\ldots ,e_q\) positive integers, then neither \(\alpha_1,\ldots ,\alpha_q\), nor any of the quotients \(\alpha_i/\alpha_j\) \((1\leq i<j\leq q)\), is a root of unity. Denoting by \(N_U(a)\) the number of integers \(m\in{\mathbb{Z}}\) with \(u_m=a\), a theorem of Skolem, Mahler and Lech states that for every every algebraic number \(a\), \(N_U(a)\) is finite. According to a conjecture of Ward, there is a uniform bound \(C(n)\) depending only on \(n\) (the order of the recurrence sequence) such that for all \(a\) we have \(N_U(a)\leq C(n)\). The author makes an important step towards proving Ward's conjecture by proving the following result: Suppose that all terms of \(U\) are contained in an algebraic number field \(K\) of degree \(d\). Then for every \(a\in K\) one has \[ N_U(a)\leq d^{6(n+1)^2}2^{2^{28(n+1)!}}. \] This implies Ward's conjecture for rational recurrence sequences. For arbitrary algebraic sequences, Ward's conjecture does not follow because of the dependence of the upper bound on \(d\). This new result of the author improves upon an earlier one, in which the upper bound for \(N_U(a)\) depended also on the number of prime ideals occurring in the prime ideal decompositions of the zeros \(\alpha_1,\ldots ,\alpha_q\) of the companion polynomial \(G\). The basic tool of the author is a new, quantitative version of the so-called parametric subspace theorem, which he also proves in the paper under review. In his proof, the author uses that the term \(u_m\) of \(U\) can be expressed as \(\sum_{i=1}^q\sum_{j=0}^{e_i-1} a_{ij} m^j\alpha_i^m\) with algebraic coefficients \(a_{ij}\). By means of some determinant argument, taking \(n+1\) solutions \(m_0,\ldots ,m_n\) of \(u_m=a\), the author eliminates the coefficients \(a_{ij}\) and obtains a linear equation in \((n+1)!\) variables \(x_1+\cdots +x_{(n+1)!}=0\) in which each term is an ``exponential monomial'' in \(m_0,\ldots ,m_n\), i.e., a product of a monomial in \(m_0,\ldots ,m_n\) and certain powers of \(\alpha_1,\ldots \alpha_q\). Then he estimates the number of solutions of this linear equation using his new quantitative subspace theorem. Note by the reviewer: Meanwhile, for the case \(n=2\), the author showed that \(N_U(a)\leq C\) for some absolute constant \(C\) and \textit{F. Beukers} and \textit{H. Schlickewei} improved this to \(N_U(a)\leq 61\) [Acta Arith. 78, 189-199 (1996; Zbl 0880.11034)]. This settles Ward's conjecture for \(n=2\). Recently, the author, W.M. Schmidt, and the reviewer showed that if the companion polynomial \(G\) has only simple zeros, i.e., \(q=n\), \(e_1=\cdots =e_n=1\), then \(N_U(a)\leq \text{exp}( (n+2) \times (6n)^{4n})\). In the general case, the upper bound for \(N_U(a)\) mentioned above has been improved upon by the author and Schmidt, but still with a dependence on \(d\).