Polynomial function

A generalization of the concept of an entire rational function (see Polynomial). Let $V$ be a unitary module over an associative-commutative ring $C$ with a unit. A mapping $\phi\colon V\to C$ is called a polynomial function if $\phi=\phi_0+\dots+\phi_m$, where $\phi_i$ is a form of degree $i$ on $V$, $i=0,\dots,m$ (see Multilinear form). Most frequently, polynomial functions are considered when $V$ is a free $C$-module (for example, a vector space over a field $C$) having a finite basis $v_1,\dots,v_n$. Then the mapping $\phi\colon V\to C$ is a polynomial function if and only if $\phi(x)=F(x_1,\dots,x_n)$, where $F\in C[X_1,\dots,X_n]$ is a polynomial over $C$ and $x_1,\dots,x_n$ are the coordinates of an element $x\in V$ in the basis $v_1,\dots,v_n$. If here $C$ is an infinite integral domain, the polynomial $F$ is defined uniquely.

The polynomial functions on a module $V$ form an associative-commutative $C$-algebra $P(V)$ with a unit with respect to the natural operations. If $V$ is a free module with a finite basis over an infinite integral domain $C$, the algebra $P(V)$ is canonically isomorphic to the symmetric algebra $S(V^*)$ of the adjoint (or dual) module $V^*$, while if $V$ is a finite-dimensional vector space over a field of characteristic 0, $P(V)$ is the algebra of symmetric multilinear forms on $V$.

Comments

E.g., polynomial functions on a Banach space naturally arise when one considers Taylor approximations to a differentiable function on such a space.