A Survey on Polynomials and Polynomial and Compatible Functions

Erhard Aichinger () and Günter F. Pilz ()
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Erhard Aichinger: Johannes Kepler University Linz, Department of Algebra, Stochastics, and Knowledge Based Mathematical Systems
Günter F. Pilz: Johannes Kepler University Linz, Department of Algebra, Stochastics, and Knowledge Based Mathematical Systems

A chapter in Proceedings of the Third International Algebra Conference, 2003, pp 1-16 from Springer

Abstract: Abstract The notion of a polynomial over a commutative ring with unity is well-known. At first sight, however, it is not so clear how to define polynomials over non-commutative rings (do the “variables” have to commute with the coefficients?) and over general algebras. To every polynomial, there is a corresponding polynomial function. When is this correspondence one-to-one? Since the “variables” and the constants preserve all congruences, all polynomial functions do the same: they are “congruence compatible” functions. But when is every congruence compatible function a polynomial function? In section 1, we explain how to define polynomials and polynomial equations over general algebras, and we state several results on the solvability of such equations. In section 2, we study polynomial functions, and in section 3, we state the answers to some questions concerning polynomial and affine completeness. We focus on those results that are applicable to polynomial functions on groups, rings, vector-spaces, modules, or, more generally, on Ω-groups.

Keywords: Abelian Group; Polynomial Equation; Commutative Ring; General Algebra; Compatible Function (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-017-0337-6_1

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