Abstract

Let $K$ be a number field. It is well known that the set of recurrencesequences with entries in $K$ is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume $\lbrace a_n\rbrace $ is a recurrence sequence and suppose that all the $a_n$ have a $d^{\rm th}$ root in the field $K$; then one can choose a sequence of such $d^{\rm th}$ roots that satisfies a recurrence itself. This was proved true in a preceding paper of the second author. In this article we generalize this result to more general monic equations; the former case can be recovered for $g=X^d-Y=0$. Combining this with the Hadamard quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the monic restriction, and have a theorem that generalizes both results