Polynomial

Definition

A function $p:\mathbb{F} \rightarrow \mathbb{F}$ is called a polynomial with coefficients in $\mathbb{F}$ if there exist $a_{0},\ldots,a_{m} \in \mathbb{F}$ such that

$$ p\left. (z) \right. = a_{0} + a_{1}z + a_{2}z^{2} + \cdots + a_{m}z^{m} $$

for all $z \in \mathbb{F}$.

$\mathcal{P}\left. \left( \mathbb{F} \right) \right.$ is the set of all polynomials with coefficients in $\mathbb{F}$.

Degree of a Polynomial

Definition

A polynomial $p \in \mathcal{P}\left. \left( \mathbb{F} \right) \right.$ is said to have degree $m$ if there exist scalars $a_{0},a_{1},\ldots,a_{m} \in \mathbb{F}$ with $a_{m} \neq 0$ such that

$$ p\left. (z) \right. = a_{0} + a_{1}z + \cdots + a_{m}z^{m} $$

for all $z \in \mathbb{F}$. If $p$ has degree $m$, we write $\mathrm{\text{deg}}\left. (p) \right. = m$.

The polynomial that is identically 0 is said to have degree $- \infty$.

$\mathcal{P}_{m}\left. \left( \mathbb{F} \right) \right.$

Definition

For $m$ a non-negative integer, $\mathcal{P}_{m}\left. \left( \mathbb{F} \right) \right.$ denotes the set of all polynomials with coefficients in $\mathbb{F}$ and degree at most $m$.