Degree (mathematics)

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

Table of contents

1 Degree of polynomial
2 Degree of vertex in a graph
3 Degree of a continuous map

3.1 Simplest case, map from a circle to itself
3.2 Degree of a map between manifolds
3.3 Properties

Degree of polynomial

The degree of a term of a polynomial is the exponent on the variable in that term; the degree of polynomial is the degree of the term of highest degree. For example, the term of highest degree in 2x³ + 4x² + x + 7 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, and the degree of a polynomial is again the degree of the term of highest degree. For example, the degree of the polynomial x²y² + 3x³ + 4y is 4, the degree of the x²y² term.

Degree of vertex in a graph

In graph theory, the degree of a vertex of a graph is the number of edges incident to that vertex.

Degree of a continuous map

Simplest case, map from a circle to itself

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

There is a projection

, ,

where is the equivalence class of modulo1 (i.e. iff is an integer).

If is continuous then there exists a continuous , called a lift of to , such that . Such a lift is unique up to an additive integer constant and .

Note that is an integer and it is also continuous with respect to ; therefore the definition does not depend on choice of .

Degree of a map between manifolds

Let be a continuous map, and closed oriented -dimensional manifolds. Then the degree of is an integer such that

Here is the map induced on the dimensional homology group, and denote the fundamental classes of and .

Here is the easiest way to calculate the degree: If is smooth and is a regular value of then is a finite number of points. In a neighborhood of each the map is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If is the number of orientation preserving and is the number of orientation reversing locations, then .

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

One can a so 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 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 are homotopic if and only if deg(f) = deg(g).