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[parent] polynomial function (Definition)

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

$\displaystyle f(x) =a_0\!+\!a_1x\!+\cdots+\!a_mx^m \,\,\,\, \forall x\in R.$

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 1   If $ R$ is a commutative ring, then the set $ R/^R$ of all polynomial functions of $ R$, equipped with the operations
$\displaystyle (f\!+\!g)(x) := f(x)\!+\!g(x),\,\,\,(f\!\cdot\!g)(x) := f(x)g(x)\,\,\, \forall x\in R,$ (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

$\displaystyle f(x) = a_0\!+\!a_1x\!+\cdots+\!a_mx^m, \,\,\, g(x) = b_0\!+\!b_1x\!+\cdots+\!b_nx^n.$
Then we can give $ f\!+\!g$ by
$\displaystyle (f\!+\!g)(x) = \sum_{i=0}^k(a_i\!+\!b_i)x^i$
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
$\displaystyle (f\!\cdot\!g)(x) = a_0b_0+(a_0b_1\!+\!a_1b_0)x+(a_0b_2\!+\!a_1b_1\!+\!a_2b_0)x^2\!+\cdots+\!a_mb_nx^{m+n}$
signifies that $ f\!\cdot\!g \in R/^R$. Because also the function $ -\!f$ given by
$\displaystyle (-\!f)(x) = -\!a_0\!-\!a_1x\!-\cdots-\!a_mx^m$
and satisfying $ -\!f\!+\!f = 0:x\mapsto 0$ belongs to $ R/^R$, the subset $ R/^R$ is a subring of $ R^R$.



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See Also: concepts in set theory, sum and product and quotient of functions

Other names:  ring of polynomial functions
Also defines:  polynomial function

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Cross-references: equation, subring, operations, subset, integral domain, infinite, polynomials, field, ring, coefficients, function, commutative ring
There are 25 references to this entry.

This is version 8 of polynomial function, born on 2006-02-12, modified 2006-09-29.
Object id is 7617, canonical name is PolynomialFunction.
Accessed 3901 times total.

Classification:
AMS MSC: 13A99 (Commutative rings and algebras :: General commutative ring theory :: Miscellaneous)
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