Homogeneous function  In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. For example, a homogeneous real-valued function of two variables "x" and "y" is a real-valued function that satisfies the condition formula_1 for some constant "k" and all real numbers α. The constant "k" is called the degree of homogeneity. More generally, 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 . 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 (). The function formula_2 is homogeneous of degree 2: formula_3 For example, suppose "x" = 2, "y" = 4 and "t" = 5. Then  Any linear map is homogeneous of degree 1 since by the definition of linearity for all and . Similarly, any multilinear function is homogeneous of degree "n" since by the definition of multilinearity for all and , , ..., . It follows that the "n"-th differential of a function between two Banach spaces "X" and "Y" is homogeneous of degree "n". Monomials in "n" variables define homogeneous functions . For example, is homogeneous of degree 10 since The degree is the sum of the exponents on the variables; in this example, 10 = 5+2+3. A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions. Given a homogeneous polynomial of degree "k", it is possible to get a homogeneous function of degree 1 by raising to the power 1/"k".