# 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

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

There is a projection

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

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

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

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

### Between manifolds

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

${\displaystyle f_{m}([X])=\deg(f)[Y].\,}$

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

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

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

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 ${\displaystyle f^{-1}(p)=\{x_{1},x_{2},..,x_{n}\}\,}$ 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 ${\displaystyle f,g:S^{n}\to S^{n}\,}$ 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.