Polynomials Integer-Valued on Maximal Orders in Division Algebras

Abstract

A polynomial f(x) in Q[x] is called integer-valued if f(n) is in Z for all n in Z. Bhargava's p-orderings and p-sequences have been helpful tools in the study of integer-valued polynomials over subsets of Z and arbitrary Dedekind domains, and similar useful definitions exist of nu-orderings and nu-sequences in the case of certain noncommutative rings. In a 2015 paper by Evrard and Johnson, these nu-sequences are used to construct a regular p-local basis for the rational integer-valued polynomials over the ring of 2x2 integer matrices M_2(Z) by way of moving the problem to maximal orders within an index 2 division algebra over Q_p. In this work, we will demonstrate how the construction used there extends nicely to maximal orders in index p division algebras over Q_2, where p is an odd prime, thereby giving the construction for a regular basis for polynomials that are integer-valued over this maximal order