Degree (mathematics)

From Exampleproblems

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

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

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. $x ˜ y$ 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).