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Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. [1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.
In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. [1]: 146 For example, in an economy with two goods ,, homothetic preferences can be represented by a utility function that has the following property: for every >:
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written (,) = (,), where f and g are homogeneous functions of the same degree of x and y. [1] In this case, the change of variable y = ux leads to an equation of the form
For a given n the elements of are then called homogeneous elements of degree n. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.
Any distribution S on R homogeneous of degree α ≠ −1, −2, ... is of this form as well. As a result, every homogeneous distribution of degree α ≠ −1, −2, ... on R \\ {0} extends to R. Finally, homogeneous distributions of degree −k, a negative integer, on R are all of the form:
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g 1, ..., g n of degree 1, then the map which sends X i onto g i defines an homomorphism of graded rings from = [, …,] onto S.
Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1 (equivalently, plethystic exponentials satisfy the usual properties of an exponential), and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of t m.