nLab polynomial function

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 Definition

In commutative rings

Without scalar coefficients

Let RR be a commutative ring. A polynomial function is a a function f:RRf:R \to R such that

  • ff is in the image of the function j:R *(RR)j:R^* \to (R \to R) from the free monoid R *R^* on RR, i.e. the set of lists of elements in RR, to the function algebra RRR \to R, such that

    • j(ϵ)=0j(\epsilon) = 0, where 00 is the zero function.
    • for all aR *a \in R^* and bR *b \in R^*, j(ab)=j(a)+j(b)() len(a)j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)}, where () n(-)^n is the nn-th power function for nn \in \mathbb{N}
    • for all rRr \in R, j(r)=c rj(r) = c_r, where c rc_r is the constant function whose value is always rr.
  • ff is in the image of the canonical ring homomorphism i:R[x](RR)i:R[x] \to (R \to R) from the polynomial ring in one indeterminant R[x]R[x] to the function algebra RRR \to R, which takes constant polynomials in R[x]R[x] to constant functions in RRR \to R and the indeterminant xx in R[x]R[x] to the identity function id R\mathrm{id}_R in RRR \to R

With scalar coefficients

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.

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 August 21, 2024 at 01:48:32. See the history of this page for a list of all contributions to it.