nLab homogeneous function

Definition

For $R$ some ring and $N,N_1,\dots,N_n$ modules over this ring, a function

$$f \;\colon\; N_1 \times \cdots \times N_n \longrightarrow N$$

is homogeneous of degree $k$ if for all $r \in R$ we have

$$f(r a_1, \cdots, r a_n) = r^k f(a_1, \cdots, a_n) \,.$$

Examples

• quadratic form

Properties

See Euler's theorem on homogeneous functions.

References

• Richard Courant: Homogeneous Functions, pp. 108 in vol 1 of: Differential & Integral Calculus, 2 volumes (1936) [vol1: ISBN:978-1-118-03149-0, ark:/13960/t4fn6m190, vol2: ISBN:978-0-471-60840-0, ark:/13960/t2g82wq0c, pdf]

See also

• Wikipedia, Homogeneous function