nLab polynomial function

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 Definition

In commutative rings

Elementary defintion

For a commutative ring RR, a polynomial function is a function f:RRf:R \to R with a natural number nn \in \mathbb{N} and a function a:[0,n]Ra:[0, n] \to R from the set of natural numbers less than or equal to nn to RR, such that for all xRx \in R,

f(x)= i:[0,n]a(i)x if(x) = \sum_{i:[0, n]} a(i) \cdot x^i

where x ix^i is the ii-th power function for multiplication.

Structural definition

For a commutative ring RR, let R[𝟙]R[\mathbb{1}] be the free commutative R R -algebra on the singleton 𝟙\mathbb{1} with element 0𝟙0 \in \mathbb{1}, with canonical function x:𝟙R[𝟙]x:\mathbb{1} \to R[\mathbb{1}], and let RRR \to R be the function algebra of RR, with identity function id R:RRid_R:R \to R. Since both objects are commutative RR-algebras there are canonical commutative ring homomorphisms α:RR[x]\alpha:R \to R[x] and β:R(RR)\beta:R \to (R \to R), and there exists a commutative ring homomorphism i:R[𝟙](RR)i:R[\mathbb{1}] \to (R \to R) such that iα=βi \circ \alpha = \beta and i(x(0))=id Ri(x(0)) = id_R. A polynomial function is a function f:RRf:R \to R in the image of i:R[𝟙](RR)i:R[\mathbb{1}] \to (R \to R).

In non-commutative algebras

For a commutative ring RR and a RR-non-commutative algebra AA, a RR-polynomial function is a function f:AAf:A \to A with a natural number nn \in \mathbb{N} and a function a:[0,n]Ra:[0, n] \to R from the set of natural numbers less than or equal to nn to RR, such that for all xAx \in A,

f(x)= i:[0,n]a(i)x if(x) = \sum_{i:[0, n]} a(i) x^i

where x ix^i is the ii-th power function for the (non-commutative) multiplication.

See also

References

Last revised on June 2, 2022 at 07:42:31. See the history of this page for a list of all contributions to it.