Arithmetic properties of certain recurrence sequences

Abstract

A classical theorem states that if a polynomial with integral coefficients is an mth power for every integral value of its argument, then it is the mth power of a polynomial with integral coefficients.In this paper we deal with analogous problems concerning functions which arise as solutions of recurrence equations with constant coefficients.

A Note On the Hadamard Kth Root of a Rational Function

“…Our result generalises that of Pisot [8] in which the unique pole must be minimal, and of multiplicity one (and the given sequence of kth roots (a' h ) is a sequence of rational integers); see also the remarks in [4]. Perelli and Zannier [7] show that the restriction on multiplicity may be removed, but they require the /?, to be positive rational integers; so unique minimality becomes trivial. The allegation in [11] that Pisot's conjecture is accessible in general is quite unfounded.…”

“…The gist of the argument below is that in a generalised power sum with polynomial coefficients, the coefficients may be shown to behave independently of the pure exponential terms. It is a generalisation of a proof of Perelli and Zannier [7], pages 13-15. It has been subsequently drawn to our attention that Proposition 3 was proved by J. P. Bezivin [3] by a different />-adic method.…”

“…and such that b(h) has constant coefficients (and takes values in K k ). (7 ] PROOF. The gist of the argument below is that in a generalised power sum with polynomial coefficients, the coefficients may be shown to behave independently of the pure exponential terms.…”

“…Perelli and Zannier [7] show that the restriction on multiplicity may be removed, but they require the /?, to be positive rational integers; so unique minimality becomes trivial. The allegation in [11] that Pisot's conjecture is accessible in general is quite unfounded.…”