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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.
The rational function defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its cone of definition is the linear cone of the points where the value of denominator is not zero.
More concretely, an n-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K: (, …,) = = =,. This formula may be rewritten using matrices: let x be the column vector with components x 1 , ..., x n and A = ( a ij ) be the n × n matrix over K whose entries are the coefficients of q .
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.
However, because f is homogeneous, meaning that f (λx 0, ..., λx n) = λ d f (x 0, ..., x n), it does make sense to ask whether f vanishes at a point [x 0 : ... : x n]. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in P n on which the functions in S vanish:
In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. [c] For example, x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5. For more details, see ...
The restriction on the characteristic is needed because otherwise a common zero of the partial derivative is not necessarily a zero of the polynomial (see Euler's identity for homogeneous polynomials). In the case of a homogeneous bivariate polynomial of degree d, this general discriminant is times the discriminant defined in § Homogeneous ...
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal.