# polynomial function

Definition.  Let $R$ be a commutative ring.  A function$f:R\to R$  is called a polynomial function of $R$, if there are some elements$a_{0},\,a_{1},\,\ldots,\,a_{m}$ of $R$ such that

 $f(x)\;=\;a_{0}\!+\!a_{1}x\!+\cdots+\!a_{m}x^{m}\,\,\,\,\forall x\in R.$

Remark.  The coefficients $a_{i}$ in a polynomial function need not be unique; e.g. if  $R=\{0,\,1\}$  is the ring (and field) of two elements, then the polynomials $X$ and $X^{2}$ both may be used for the same polynomial function.  However, if we stipulate that $R$ is an infinite integral domain, the coefficients are guaranteed to be unique.

The set of all polynomial functions of $R$, being a subset of the set $R^{R}$ of all functions from $R$ to $R$, is here denoted by  $R/^{R}$.

###### Theorem.

If $R$ is a commutative ring, then the set $R/^{R}$ of all polynomial functions of $R$, equipped with the operations

 $\displaystyle(f\!+\!g)(x)\;:=\;f(x)\!+\!g(x),\quad(f\!\cdot\!g)(x)\;:=\;f(x)g(% x)\quad\forall x\in R,$ (1)

is a commutative ring.

Proof.  It’s straightforward to show that the function set $R^{R}$ forms a commutative ring when equipped with the operations “$+$” and “$\cdot$” defined as (1).  We show now that $R/^{R}$ forms a subring of $R^{R}$.  Let $f$ and $g$ be any two polynomial functions given by

 $f(x)\;=\;a_{0}\!+\!a_{1}x\!+\cdots+\!a_{m}x^{m},\,\,\,g(x)\;=\;b_{0}\!+\!b_{1}% x\!+\cdots+\!b_{n}x^{n}.$

Then we can give $f\!+\!g$ by

 $(f\!+\!g)(x)\;=\;\sum_{i=0}^{k}(a_{i}\!+\!b_{i})x^{i}$

where  $k=\max\{m,\,n\}$  and  $a_{i}=0$ (resp.  $b_{i}=0$) for  $i>m$ (resp.  $i>n$).  This means that  $f\!+\!g\in R/^{R}$.  Secondly, the equation

 $(f\!\cdot\!g)(x)\;=\;a_{0}b_{0}+(a_{0}b_{1}\!+\!a_{1}b_{0})x+(a_{0}b_{2}\!+\!a% _{1}b_{1}\!+\!a_{2}b_{0})x^{2}\!+\cdots+\!a_{m}b_{n}x^{m+n}$

signifies that  $f\!\cdot\!g\in R/^{R}$.  Because also the function $-\!f$ given by

 $(-\!f)(x)\;=\;-\!a_{0}\!-\!a_{1}x\!-\cdots-\!a_{m}x^{m}$

and satisfying  $-\!f\!+\!f=0:x\mapsto 0$  belongs to $R/^{R}$, the subset $R/^{R}$ is a subring of $R^{R}$.

 Title polynomial function Canonical name PolynomialFunction Date of creation 2013-03-22 15:40:34 Last modified on 2013-03-22 15:40:34 Owner pahio (2872) Last modified by pahio (2872) Numerical id 13 Author pahio (2872) Entry type Definition Classification msc 13A99 Synonym ring of polynomial functions Related topic NotationInSetTheory Related topic ProductAndQuotientOfFunctionsSum Related topic ZeroOfPolynomial Related topic PolynomialFunctionIsAProperMap Related topic DerivativeOfPolynomial Defines polynomial function