Polynomial functions on upper triangular matrix algebras

TL;DR

This paper characterizes polynomial functions on upper triangular matrix algebras over commutative rings, showing their structure and properties related to integer-valued and null-polynomials, with implications for algebraic ring theory.

Contribution

It provides a characterization of polynomial functions induced by algebraic and scalar polynomials on upper triangular matrices, and analyzes their algebraic properties.

Findings

Integer-valued polynomials form a ring.

Null-polynomials form an ideal.

Characterization of polynomial functions in terms of substitution homomorphism.

Abstract

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.