# Degree (mathematics)

In mathematics, there are several meanings of degree depending on the subject.

## Degree of a polynomial

See main article Degree of a polynomial

The degree of a term of a polynomial in one variable is the exponent on the variable in that term; the degree of a polynomial is the highest such degree. For example, in 2x3 + 4x2 + x + 7, the term of highest degree is 2x3; this term, and therefore the entire polynomial, are said to have degree 3.

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 of the polynomial is again the highest such degree. For example, the polynomial x2y2 + 3x3 + 4y has degree 4, the same degree as the term x2y2.

## Degree of a vertex in a graph

See main article degree (graph theory)

In graph theory, the degree of a vertex in a graph is the number of edges incident to that vertex — in other words, the number of lines coming out of the point.

## Degree of a continuous map

See main article degree (continuous map)

In topology, the term degree is applied to continuous maps between manifolds of the same dimension.

### From a circle to itself

The simplest and most important case is the degree of a continuous map

$f\colon S^{1}\to S^{1}\,$ .

There is a projection

$\mathbb {R} \to S^{1}=\mathbb {R} /\mathbb {Z} \,$ , $x\mapsto [x]$ ,

where $[x]$ is the equivalence class of $x$ modulo 1 (i.e. $x\sim y$ iff $x-y$ is an integer).

If $f:S^{1}\to S^{1}\,$ is continuous then there exists a continuous $F:\mathbb {R} \to \mathbb {R}$ , called a lift of $f$ to $\mathbb {R}$ , such that $f([z])=[F(z)]\,$ . Such a lift is unique up to an additive integer constant and $deg(f)=F(x+1)-F(x)\,$ .

Note that $F(x+1)-F(x)$ is an integer and it is also continuous with respect to $x$ ; therefore the definition does not depend on choice of $x$ .

### Between manifolds

Let $f:X\to Y\,$ be a continuous map, $X$ and $Y$ closed oriented $m$ -dimensional manifolds. Then the degree of $f$ is an integer such that

$f_{m}([X])=\deg(f)[Y].\,$ Here $f_{m}$ is the map induced on the $m$ dimensional homology group, $[X]$ and $[Y]$ denote the fundamental classes of $X$ and $Y$ .

Here is the easiest way to calculate the degree: If $f$ is smooth and $p$ is a regular value of $f$ then $f^{-1}(p)=\{x_{1},x_{2},..,x_{n}\}\,$ is a finite number of points. In a neighborhood of each the map $f$ is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If $m$ is the number of orientation preserving and $k$ is the number of orientation reversing locations, then $deg(f)=m-k\,$ .

The same definition works for compact manifolds with boundary but then $f$ should send the boundary of $X$ to the boundary of $Y$ .

One can also define degree modulo 2 (deg2(f)) the same way as before but taking the fundamental class in Z2 homology. In this case deg2(f) is element of Z2, the manifolds need not be orientable and if $f^{-1}(p)=\{x_{1},x_{2},..,x_{n}\}\,$ as before then deg2(f) is n modulo 2.

### Properties

The degree of map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps $f,g:S^{n}\to S^{n}\,$ are homotopic if and only if deg(f) = deg(g).

## Degree of freedom

A degree of freedom is a concept in mathematics, statistics, physics and engineering. See degrees of freedom.