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polynomial function
Contents
Context
Algebra
algebra , higher algebra
universal algebra
monoid , semigroup , quasigroup
nonassociative algebra
associative unital algebra
commutative algebra
Lie algebra , Jordan algebra
Leibniz algebra , pre-Lie algebra
Poisson algebra , Frobenius algebra
lattice , frame , quantale
Boolean ring , Heyting algebra
commutator , center
monad , comonad
distributive law
Group theory
Ring theory
Module theory
Contents
Definition
In commutative rings
Without scalar coefficients
Let R R be a commutative ring . A polynomial function is a a function f : R → R f:R \to R such that
f f is in the image of the function j : R * → ( R → R ) j:R^* \to (R \to R) from the free monoid R * R^* on R R , i.e. the set of lists of elements in R R , to the function algebra R → R R \to R , such that
j ( ϵ ) = 0 j(\epsilon) = 0 , where 0 0 is the zero function.
for all a ∈ R * a \in R^* and b ∈ R * b \in R^* , j ( a b ) = j ( a ) + j ( b ) ⋅ ( − ) len ( a ) j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)} , where ( − ) n (-)^n is the n n -th power function for n ∈ ℕ n \in \mathbb{N}
for all r ∈ R r \in R , j ( r ) = c r j(r) = c_r , where c r c_r is the constant function whose value is always r r .
f f 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 → R R \to R , which takes constant polynomials in R [ x ] R[x] to constant functions in R → R R \to R and the indeterminant x x in R [ x ] R[x] to the identity function id R \mathrm{id}_R in R → R R \to R
With scalar coefficients
For a commutative ring R R , a polynomial function is a function f : R → R f:R \to R with a natural number n ∈ ℕ n \in \mathbb{N} and a function a : [ 0 , n ] → R a:[0, n] \to R from the set of natural numbers less than or equal to n n to R R , such that for all x ∈ R x \in R ,
f ( x ) = ∑ i : [ 0 , n ] a ( i ) ⋅ x i f(x) = \sum_{i:[0, n]} a(i) \cdot x^i
where x i x^i is the i i -th power function for multiplication.
In non-commutative algebras
For a commutative ring R R and a R R -non-commutative algebra A A , a R R -polynomial function is a function f : A → A f:A \to A with a natural number n ∈ ℕ n \in \mathbb{N} and a function a : [ 0 , n ] → R a:[0, n] \to R from the set of natural numbers less than or equal to n n to R R , such that for all x ∈ A x \in A ,
f ( x ) = ∑ i : [ 0 , n ] a ( i ) x i f(x) = \sum_{i:[0, n]} a(i) x^i
where x i x^i is the i i -th power function for the (non-commutative) multiplication.
See also
References
Last revised on August 21, 2024 at 01:48:32.
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