Homogeneous function

of degree $ \lambda $

A function $ f $ such that for all points $ ( x _ {1} \dots x _ {n} ) $ in its domain of definition and all real $ t > 0 $, the equation

$$ f ( t x _ {1} \dots t x _ {n} ) = \ t ^ \lambda f ( x _ {1} \dots x _ {n} ) $$

holds, where $ \lambda $ is a real number; here it is assumed that for every point $ ( x _ {1} \dots x _ {n} ) $ in the domain of $ f $, the point $ ( t x _ {1} \dots t x _ {n} ) $ also belongs to this domain for any $ t > 0 $. If

$$ f ( x _ {1} \dots x _ {n} ) = \ \sum _ {0 \leq k _ {1} + \dots + k _ {n} \leq m } a _ {k _ {1} \dots k _ {n} } x _ {1} ^ {k _ {1} } \dots x _ {n} ^ {k _ {n} } , $$

that is, $ f $ is a polynomial of degree not exceeding $ m $, then $ f $ is a homogeneous function of degree $ m $ if and only if all the coefficients $ a _ {k _ {1} \dots k _ {n} } $ are zero for $ k _ {1} + \dots + k _ {n} < m $. The concept of a homogeneous function can be extended to polynomials in $ n $ variables over an arbitrary commutative ring with an identity.

Suppose that the domain of definition $ E $ of $ f $ lies in the first quadrant, $ x _ {1} > 0 \dots x _ {n} > 0 $, and contains the whole ray $ ( t x _ {1} \dots t x _ {n} ) $, $ t > 0 $, whenever it contains $ ( x _ {1} \dots x _ {n} ) $. Then $ f $ is homogeneous of degree $ \lambda $ if and only if there exists a function $ \phi $ of $ n- 1 $ variables, defined on the set of points of the form $ ( x _ {2} / x _ {1} \dots x _ {n} / x _ {1} ) $ where $ ( x _ {1} \dots x _ {n} ) \in E $, such that for all $ ( x _ {1} \dots x _ {n} ) \in E $,

$$ f ( x _ {1} \dots x _ {n} ) = \ x _ {1} ^ \lambda \phi \left ( { \frac{x _ 2}{x _ 1} } \dots { \frac{x _ n}{x _ 1} } \right ) . $$

If the domain of definition $ E $ of $ f $ is an open set and $ f $ is continuously differentiable on $ E $, then the function is homogeneous of degree $ \lambda $ if and only if for all $ ( x _ {1} \dots x _ {n} ) $ in its domain of definition it satisfies the Euler formula

$$ \sum _ { i=1 } ^ { n } x _ {i} \frac{\partial f ( x _ {1} \dots x _ {n} ) }{\partial x _ {i} } = \ \lambda f ( x _ {1} \dots x _ {n} ) . $$