What is "homogeneous function"

n. 1 (context mathematics English) homogeneous polynomial 2 (context mathematics English) the ratio of two homogeneous polynomials, such that the sum of the exponents in a term of the numerator is equal to the sum of the exponents in a term of the denominator. 3 (context mathematics English) a function ''f''(''x'') which has the property that for any ''c'', $f(c\; x)\; =\; c\; f(x)$.

In mathematics, a homogeneous function is a function which satisfies the condition f(tx, ty) = tf(x, y), for some integer n.

It can also be described as a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if for all nonzero and . This implies it has scale invariance. When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that hold for all α > 0.

Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by .