Degree of a polynomial

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.[1][2] The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see order of a polynomial (disambiguation)).

The following names are assigned to polynomials according to their degree:[3][4][5][2]

For higher degrees, names have sometimes been proposed,[7] but they are rarely used:

The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.[8]

The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; that is,

The degree of the product of a polynomial by a non-zero scalar is equal to the degree of the polynomial; that is,

Thus, the set of polynomials (with coefficients from a given field F) whose degrees are smaller than or equal to a given number n forms a vector space; for more, see Examples of vector spaces.

More generally, the degree of the product of two polynomials over a field or an integral domain is the sum of their degrees:

Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is usually undefined. The propositions for the degree of sums and products of polynomials in the above section do not apply, if any of the polynomials involved is the zero polynomial.[10]

These examples illustrate how this extension satisfies the behavior rules above:

A number of formulae exist which will evaluate the degree of a polynomial function f. One based on asymptotic analysis is

this is the exact counterpart of the method of estimating the slope in a log–log plot.

This formula generalizes the concept of degree to some functions that are not polynomials. For example:

For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x2y2 + 3x3 + 4y has degree 4, the same degree as the term x2y2.

However, a polynomial in variables x and y, is a polynomial in x with coefficients which are polynomials in y, and also a polynomial in y with coefficients which are polynomials in x. The polynomial

Given a ring R, the polynomial ring R[x] is the set of all polynomials in x that have coefficients in R. In the special case that R is also a field, the polynomial ring R[x] is a principal ideal domain and, more importantly to our discussion here, a Euclidean domain.

It can be shown that the degree of a polynomial over a field satisfies all of the requirements of the norm function in the euclidean domain. That is, given two polynomials f(x) and g(x), the degree of the product f(x)g(x) must be larger than both the degrees of f and g individually. In fact, something stronger holds:

Since the norm function is not defined for the zero element of the ring, we consider the degree of the polynomial f(x) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.