The notion of a polynomial over a commutative ring with identity is well known. It is not so clear how to define polynomials over general rings (when do "indeterminates" commute?,...) and over other algebraic systems. This can be done in the context of universal algebra: the polynomial algebra over an algebraic structure A in a variety V with the set X of indeterminates is the free union of A and the free algebra over X in V. To every polynomial, there is a corresponding polynomial function. The collection of polynomial functions over A in X can be defined as the subalgebra of all functions from A to A, generated by the projections and the constant maps. An open question is the charactderization, when there is a 1-1 corrsspondence between polynomials and polynomial functions. These considerations have a remarkable impact on questions concerning algebraic equations over algebras.
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