Pisot numbers, Salem numbers, and generalised polynomials

TL;DR

This paper explores the relationship between generalised polynomial expressions and special integer sets generated by linear recurrent sequences of Salem and Pisot types, introducing new concepts in number field analysis.

Contribution

It introduces the notion of generalised polynomials on number fields and links their existence to the structure of linear recurrent sequences and subsemigroups in number fields.

Findings

Sets of values of Salem-type sequences can be described by generalised polynomials.

Some Pisot-type sequence values are also representable by generalised polynomials.

A connection is established between generalised polynomial expressions and subsemigroups in number fields.

Abstract

We study sets of integers that can be defined by the vanishing of a generalised polynomial expression. We show that this includes sets of values of linear recurrent sequences of Salem type and some linear recurrent sequences of Pisot type. To this end, we introduce the notion of a generalised polynomial on a number field. We establish a connection between the existence of generalised polynomial expressions for sets of values of linear recurrent sequences and for subsemigroups of multiplicative groups of number fields.