Polynomial

Definition: Polynomial, $\mathcal{P}(\mathbb{F})$

A function $p:\mathbb{F}\to \mathbb{F}$ is called a polynomial with coefficients in $\mathbb{F}$ if there exists $a_0,\dots ,a_m\in \mathbb{F}$ such that

$$p(z)=a_0+a_{1}z+a_2{z}^2+\cdots +a_m{z}^m$$

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

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

With the usual operations of addition and scalar multiplication, $\mathcal{P}(\mathbb{F})$ is a vector space of $\mathbb{F}$. Hence, $\mathcal{P}(\mathbb{F})$ is a subspace of ${\mathbb{F}}^{\mathbb{F}}$, the vector space of functions from $\mathbb{F}$ to $\mathbb{F}$.

If a polynomial (thought of as a function from $\mathbb{F}$ to $\mathbb{F}$) is represented by two sets of coefficients, then subtracting one representation of the polynomial from the other produces a polynomial that is identically zero as a function on $\mathbb{F}$ and hence has all zero coefficients. Thus, the coefficients of a polynomial are uniquely determined by the polynomial.

Definition: Degree of a polynomial

A polynomial $p\in \mathcal{P}(\mathbb{F})$ is said to have degree $m$ if there exist scalars $a_0,a_1,\dots a_m\in \mathbb{F}$ with $a_m\ne 0$ such that for every $z\in \mathbb{F}$, we have

$$p(z)=a_0+a_{1}z+\cdots +a_m{z}^m$$

• The polynomial that is identically $0$ is said to have degree $-\infty$
• The degree of a polynomial $p$ is denoted by $\mathrm{deg}p$

Notation: ${\mathcal{P}}_m(\mathbb{F})$

For $m$, a nonnegative integer, ${\mathcal{P}}_m(\mathbb{F})$ denotes the set of all polynomials with coefficients in $\mathbb{F}$ and degree at most $m$.

If $m$ is a nonnegative integer, ${\mathcal{P}}_m(\mathbb{F})=\text{span}(1,z,\dots ,z^m)$. Here, we are letting $z^k$ denote a function.

Thus, ${\mathcal{P}}_m(\mathbb{F})$ is a finite-dimensional vector space for each nonnegative integer $m$.

On the other hand, $\mathcal{P}(\mathbb{F})$ is infinite-dimensional. Consider any list of elements of $\mathcal{P}(\mathbb{F})$. Let $m$ denote the highest degree of the polynomials in this list. Then, every polynomial in the span of this list has degree at most $m$. Then, $z^{m+1}$ is not in the span of our list; therefore, the span of any finite list of polynomials cannot span the entire $\mathcal{P}(\mathbb{F})$. Thus, $\mathcal{P}(\mathbb{F})$ is infinite-dimensional.