polynomial function
Definition. Let be a commutative ring. A function is called a polynomial function of , if there are some elements of such that
Remark. The coefficients in a polynomial function need not be unique; e.g. if is the ring (and field) of two elements, then the polynomials and both may be used for the same polynomial function. However, if we stipulate that is an infinite integral domain
, the coefficients are guaranteed to be unique.
The set of all polynomial functions of , being a subset of the set of all functions from to , is here denoted by .
Theorem.
If is a commutative ring, then the set of all polynomial functions of , equipped with the operations
(1) |
is a commutative ring.
Proof. It’s straightforward to show that the function set forms a commutative ring when equipped with the operations “” and “” defined as (1). We show now that forms a subring of . Let and be any two polynomial functions given by
Then we can give by
where and (resp. ) for (resp. ). This means that . Secondly, the equation
signifies that . Because also the function given by
and satisfying belongs to , the subset is a subring of .
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 |