polynomial function (changes) in nLab

Showing changes from revision #5 to #6: Added | Removed | Changed

Contents

 Definition

In commutative rings

Without scalar coefficients

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

• ff is in the image of the function j:R *→(R→R)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 R→RR \to R, such that

  • j(ϵ)=0j(\epsilon) = 0, where 00 is the zero function.
  • for all a∈R *a \in R^* and b∈R *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 n∈ℕn \in \mathbb{N}
  • for all r∈Rr \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]→(R→R)i:R[x] \to (R \to R) from the polynomial ring in one indeterminant R[x]R[x] to the function algebra R→RR \to R, which takes constant polynomials in R[x]R[x] to constant functions in R→RR \to R and the indeterminant xx in R[x]R[x] to the identity function id R\mathrm{id}_R in R→RR \to R

With scalar coefficients

For a commutative ring RR, a polynomial function is a function f:R→Rf:R \to R with a natural number n∈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 x∈Rx \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:A→Af:A \to A with a natural number n∈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 x∈Ax \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

• real polynomial function

• quadratic function

• rational zero theorem

• fundamental theorem of algebra

• polynomial

• degree of a polynomial

References

• Wikipedia, Polynomial function