A monomial is a product of non-negative powers of variables. It may also include an optional coefficient (which is sometimes ignored when discussing particular properties of monomials). A polynomial can be thought of as a sum over a set of monomials.

For example, the following are monomials.

$$ \begin{array}{ccc} 1 & x & x^2y \\ \\ xyz & 3x^4y^2z^3 & -z \end{array} $$

If there are $n$ variables from which a monomial may be formed, then a monomial may be represented without its coefficient as a vector of $n$ naturals. Each position in this vector would correspond to a particular variable, and the value of the element at each position would correspond to the power of that variable in the monomial. For instance, the monomial $x^2yz^3$ formed from the set of variables $\left\{ w, x, y, z \right\}$ would be represented as $\begin{pmatrix}0&2&1&3\end{pmatrix}^T$ A constant would be a zero vector.

Given this representation, we may define a few more concepts. First, the degree of a monomial is the sum of the elements of its vector representation. Thus, the degree of $x^2yz^3$ is $0 + 2 + 1 + 3 = 6$ and the degree of a constant is 0. If a polynomial is represented as a sum over a set of monomials, then the degree of a polynomial can be defined as the degree of the monomial of largest degree belonging to that polynomial.