In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example...

elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...

to a kth-degree or kth-order homogeneous function. For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above...

quadratic form; and more generally, the discriminant of a form, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially...

and drawing of curves defined by a bivariate polynomial equation. The resultant of n homogeneous polynomials in n variables (also called multivariate resultant...

weight or the degree of the polynomial. The term quasi-homogeneous comes from the fact that a polynomial f is quasi-homogeneous if and only if f ( λ w 1...

complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression...

Diophantine equations. A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial. A typical such equation...

^{k}f(x,y,z)=0.} A polynomial g ( x , y ) {\displaystyle g(x,y)} of degree k {\displaystyle k} can be turned into a homogeneous polynomial by replacing x...

quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered...

elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible...

The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, x3y2 + 7x2y3 − 3x5 is homogeneous of degree...

{\displaystyle R_{i}} consisting of homogeneous polynomials of degree i. Let S be the set of all nonzero homogeneous elements in a graded integral domain...

set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables...

P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

projective Nullstellensatz states that, for any homogeneous ideal I that does not contain all polynomials of a certain degree (referred to as an irrelevant...

defined by homogeneous polynomials in n + 1 indeterminates, then N is either infinite, or equals the product of the degrees of the polynomials. Moreover...

In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only...

a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play...

Greatest common divisior of two polynomials Symmetric function Homogeneous polynomial Polynomial SOS (sum of squares) Polynomial family Quadratic function Cubic...

in k[x0, ..., xn] be a homogeneous polynomial of degree d. It is not well-defined to evaluate  f  on points in Pn in homogeneous coordinates. However,...

and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the...

Homogeneous linear transformation Homogeneous model in model theory Homogeneous polynomial Homogeneous relation: binary relation on a set Homogeneous...

mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y...

the polynomial that follows it. The dimension of the vector space K [ x 1 , … , x n ] d {\displaystyle K[x_{1},\ldots ,x_{n}]_{d}} of homogeneous polynomial...

"spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions R 3 → R {\displaystyle \mathbb {R} ^{3}\to \mathbb {R}...

needed] In algebra, homogeneous polynomials have the same number of factors of a given kind. In the study of binary relations, a homogeneous relation R is on...

function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. The equation...

K[X0, X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective...