polynomial function

Definition.  Let R be a commutative ring.  A functionf:RR  is called a polynomial function of R, if there are some elementsa0,a1,,am of R such that


Remark.  The coefficients ai 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 X2 both may be used for the same polynomial function.  However, if we stipulate that R is an infiniteMathworldPlanetmathPlanetmath integral domainMathworldPlanetmath, the coefficients are guaranteed to be unique.

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


If R is a commutative ring, then the set R/R of all polynomial functions of R, equipped with the operationsMathworldPlanetmath

(f+g)(x):=f(x)+g(x),(fg)(x):=f(x)g(x)xR, (1)

is a commutative ring.

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


Then we can give f+g by


where  k=max{m,n}  and  ai=0 (resp.  bi=0) for  i>m (resp.  i>n).  This means that  f+gR/R.  Secondly, the equation


signifies that  fgR/R.  Because also the function -f given by


and satisfying  -f+f=0:x0  belongs to R/R, the subset R/R is a subring of RR.

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