Definition:Polynomial
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Definition
Let $R$ be a commutative ring.
Let $D$ be a subring of $R$.
A polynomial in $x$ over $R$ is an element of the form:
- $P \left({x}\right) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_a x + a_0$
for some $n \ge 0$ where $x \in R$ and each of the $a_j$ are elements of $D$.
Polynomial Form
Let $M$ be the free commutative monoid on the indexed set $\left\{{X_j: j \in J}\right\}$.
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity with additive identity $0_R$ and multiplicative identity $1_R$.
A polynomial form or just polynomial over $\left\{{X_j: j \in J}\right\}$ is a mapping $f: M \to R: \mathbf X^k \mapsto a_k$ such that $a_k = 0$ for all but finitely many $\mathbf X^k \in M$.
Indeterminates
The elements of the set $\left\{{X_j: j \in J}\right\}$ are called indeterminates.
If $\left\{{X_j: j \in J}\right\} = \left\{{X}\right\}$ is a singleton, then the indeterminate $\left\{{X}\right\}$ is often unimportant, and we speak of the polynomial $f$ over the ring $R$.
Polynomial Function
Let $K$ be a commutative ring with unity.
Let the mapping $p: K \to K$ be defined such that there exists a sequence:
- $\left \langle {\alpha_k} \right \rangle_{k \in \left[{0 \,.\,.\, n}\right]}$
of elements of $K$ such that:
- $\displaystyle p = \sum_{k \mathop = 0}^n \alpha_k {I_K}^k$
where $I_K$ is the identity mapping on $K$.
Then $p$ is known as a polynomial function on $K$.
Polynomial Function: General Definition
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$.
For each $x = \left({x_j}\right)_{j \in J} \in R^J$, let $\phi_x: R \left[\left\{{X_j: j \in J}\right\}\right] \to R$ be the evaluation homomorphism from the ring of polynomial forms at $x$.
Then the set:
- $\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in R^J}\right\} \subseteq R^J \times R$
defines a polynomial function $R^J \to R$.
Polynomial Equation
A polynomial equation is an equation in the form:
- $f \left({x}\right) = 0$
where $f$ is a polynomial function.
Degree
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{{X_j: j \in J}\right\}$ that is not the null polynomial for some multiindices $k_1, \ldots, k_r$.
For a multiindex $k = \left({k_j}\right)_{j \mathop \in J}$, let $\displaystyle \left|{k}\right| = \sum_{j \mathop \in J} k_j \ge 0$ be the degree of the mononomial $\mathbf X^k$.
The degree of $f$ is the supremum:
- $\displaystyle \deg \left({f}\right) = \max \left\{{\left| {k_r} \right|: i = 1, \ldots, r}\right\}$
Some sources write $\deg \left({f}\right)$ as $\partial f$.
The null polynomial is sometimes defined to have degree $-\infty$, but is left undefined in many sources.
Also see
Sources
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 3.11 \ (3)$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 64$
- Pierre A. Grillet: Abstract Algebra (2000): $\S \text{III}.6$
