monomial

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^{2}y\\\\\\\\xyz&3x^{4}y^{2}z^{3}&-z\\end{array}

If there are $n$ variables from which a monomial may be formed, thena monomial may be represented without its coefficient as a vector of $n$ naturals. Each position in this vector would correspond to a particularvariable, and the value of the element at each position would correspondto the power of that variable in the monomial. For instance, the monomial $x^{2}yz^{3}$ 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, thedegree of a monomial is the sum of the elements of its vector representation. Thus, the degree of $x^{2}yz^{3}$ is $0+2+1+3=6$ ,and the degree of a constant is 0. If a polynomial is represented as a sumover a set of monomials, then the degree of a polynomial can be defined as thedegree of the monomial of largest degree belonging to that polynomial.