Multiplicities of algebraic linear recurrences
Let \((u_ m)^ \infty_{m=0}\) be a recurrence sequence of order \(n\) with initial terms \(u_ 0,\dots,u_{n-1}\) defined by \[ u_{m+n}=\nu_{n-1} u_{m+n-1}+\nu_{n-2} u_{m+n-2}+ \cdots+ \nu_ 0u_ m \quad (m=0,1,2,\dots). \] The \(a\)-multiplicity \(U(a)\) is defined as the number of indices \(m\) such that \(u_ m=a\). The multiplicity \(U\) is \(U=\sup_ aU(a)\). A result of \textit{J.-H. Evertse}, \textit{K. Győry}, \textit{C. L. Stewart} and \textit{R. Tijdeman} [\(S\)-unit equations and their applications, New Advances in Trancendence Theory, Proc. Symp. Durham 1986, 110-174 (1988; Zbl 0658.10023)] implies, that a nonperiodic, nondegenerate recurrence sequence has finite multiplicity. The aim of the author is to establish an upper bound for the multiplicity of a nonperiodic, nondegenerate sequence, in case the terms \(u_ m\) of the sequence are algebraic. Let \(K\) be an algebraic number field of degree \(d\), containing the initial terms \(u_ 0,\dots,u_{n-1}\) and the coefficients \(\nu_{n- 1},\dots,\nu_ 0\) of the sequence, as well as the roots \(\alpha_ 1,\dots,\alpha_ r\) of the companion polynomial of the recurrence. Define \(\omega=\omega (\alpha_ 1,\dots,\alpha_ r)\) to be the number of prime ideals in \(K\) occurring in the decomposition of the ideals \((\alpha_ 1),\dots,(\alpha_ r)\). The main result of the paper asserts, that if the sequence is nonperiodic and nondegenerate, then \[ U \leq (4(\omega+d) d!)^{2^{\textstyle{40(n+1) !d!}} (w+d)^ 6}. \] The main tool of the proof of this deep result is the quantitative \(p\)-adic Subspace Theorem of the author [Compos. Math. 82, 245-273 (1992; Zbl 0751.11033)]. The counting method avoids to use the gap principle and applies a determinant argument instead. This argument can also be used for \(S\)-unit equations. For some related results see \textit{A. J. van der Poorten} and \textit{H. P. Schlickewei} [Bull. Aust. Math. Soc. 44, 215-223 (1991; Zbl 0724.11006)].
quantitative \(p\)-adic subspace theorem
scientific article; zbMATH DE number 469406
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