# Degree (mathematics)

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

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## 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

### From a circle to itself

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

[itex]f:S^1\to S^1 \,[itex].

There is a projection

[itex]R \to S^1=R/Z \,[itex], [itex]x\mapsto [x][itex],

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

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

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

### Between manifolds

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

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

[itex]f_m([X])=\deg(f)[Y].[itex]

Here [itex]f_m[itex] is the map induced on the [itex]m[itex] dimensional homology group, [itex][X][itex] and [itex][Y][itex] denote the fundamental classes of [itex]X[itex] and [itex]Y[itex].

Here is the easiest way to calculate the degree: If [itex]f[itex] is smooth and [itex]p[itex] is a regular value of [itex]f[itex] then [itex]f^{-1}(p)=\{x_1,x_2,..,x_n\}[itex] is a finite number of points. In a neighborhood of each the map [itex]f[itex] is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If [itex]m[itex] is the number of orientation preserving and [itex]k[itex] is the number of orientation reversing locations, then [itex]deg(f)=m-k[itex].

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

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 [itex]f^{-1}(p)=\{x_1,x_2,..,x_n\}[itex] 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 [itex]f,g:S^n\to S^n[itex] 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.

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