+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +-- {: .hide} [[!include algebra - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition ## ### In commutative rings ### #### Without scalar coefficients #### Let $R$ be a [[commutative ring]]. A **polynomial function** is a a [[function]] $f:R \to R$ such that * $f$ is in the image of the [[function]] $j:R^* \to (R \to R)$ from the [[free monoid]] $R^*$ on $R$, i.e. the set of lists of elements in $R$, to the [[function algebra]] $R \to R$, such that * $j(\epsilon) = 0$, where $0$ is the zero function. * for all $a \in R^*$ and $b \in R^*$, $j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)}$, where $(-)^n$ is the $n$-th power function for $n \in \mathbb{N}$ * for all $r \in R$, $j(r) = c_r$, where $c_r$ is the constant function whose value is always $r$. * $f$ is in the image of the canonical [[ring homomorphism]] $i:R[x] \to (R \to R)$ from the polynomial ring in one indeterminant $R[x]$ to the [[function algebra]] $R \to R$, which takes constant polynomials in $R[x]$ to constant functions in $R \to R$ and the indeterminant $x$ in $R[x]$ to the identity function $\mathrm{id}_R$ in $R \to R$ #### With scalar coefficients #### For a [[commutative ring]] $R$, a **polynomial function** is a [[function]] $f:R \to R$ with a [[natural number]] $n \in \mathbb{N}$ and a function $a:[0, n] \to R$ from the set of natural numbers less than or equal to $n$ to $R$, such that for all $x \in R$, $$f(x) = \sum_{i:[0, n]} a(i) \cdot x^i$$ where $x^i$ is the $i$-th [[power function]] for multiplication. ### In non-commutative algebras ### For a [[commutative ring]] $R$ and a $R$-[[associative unital algebra|non-commutative algebra]] $A$, a $R$-**polynomial function** is a [[function]] $f:A \to A$ with a [[natural number]] $n \in \mathbb{N}$ and a function $a:[0, n] \to R$ from the set of natural numbers less than or equal to $n$ to $R$, such that for all $x \in A$, $$f(x) = \sum_{i:[0, n]} a(i) x^i$$ where $x^i$ is the $i$-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](https://en.wikipedia.org/wiki/Polynomial_function)_ [[!redirects polynomial functions]] [[!redirects polynomial map]] [[!redirects polynomial maps]] [[!redirects polynomial mapping]] [[!redirects polynomial mappings]]