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In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous of degree k if
for every x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} and s ≠ 0. {\displaystyle s\neq 0.}
For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k.
The above definition extends to functions whose domain and codomain are vector spaces over a field F: a function f : V → W {\displaystyle f:V\to W} between two F-vector space is homogeneous of degree k {\displaystyle k} if