Expressions written as a linear combination of powers of an independent variable.
$$ p(x) = \sum _{i=0} ^d a_i x^i $$
Polynomials have two properties useful for cryptography,
$$ \textbf{Degree-Root Relation} \\ \text{A nonzero polynomial of degree $d$ has$\leq d$ roots.} $$
$$ \textbf{Interpolation Existence and Uniqueness} \\ \text{Given $(x_0, y_0), \space \dots, \space (x_d, y_d)$, there exists a unique polynomial $p$} \\ \text{with degree$\leq d$ such that $p(x_i) = y_i$ for all $i \in \{0, \space \dots , \space d\}$.} $$
Any polynomial $p$ passing through points $(x_0, y_0), \space \dots, \space (x_d, y_d)$ can be reconstructed using a linear combination of Lagrange Basis Polynomials.
$$ p(x) = \sum _{i = 0} ^d \Delta _i (x)y_i \hspace{15pt} \text{with } \Delta _i = \frac{\prod _{j \neq i}(x-x_j)}{\prod _{j \neq i} (x_i -x_j)}=\begin{cases} 1 \hspace{35pt} x=x_i \\ 0 \hspace{35pt} x=x_j \\ k \in F \hspace{12pt} \text{ otherwise } \\ \end{cases} $$
Operate over a finite set of elements $\{0, \space \dots , \space p-1 \}$ modulo $p$. If $p$ is prime, the field is said to be a Galois Field $\text{GF}(p)$ on order $p$, guaranteeing a multiplicative inverse, and ensuring that both polynomial properties hold by closure under multiplication, division, subtraction, and addition.
A polynomial $p$ is said to have a value representation and a coefficient representation.
$$ \textbf{Value Representation } \space (x_{d}, p(x_{d})) \space \dots \space (x_0, p(x_0)) \\
\textbf{Coefficient Representation } \space a_dx^d+\dots +a_0 \hspace{69pt} $$
Working over a finite field on order $k$, for any one choice of a coefficient, we have $k$ choices for the remaining $d$ coefficients, the number of unique polynomials modulo $k$ is $k^d$.