HomogenePolynome

HomogenePolynome, in the context of algebra, refers to homogeneous polynomials. A polynomial in several variables over a field k is called homogeneous of degree d if every monomial appearing in it has total degree d. Equivalently, for all t in k and all variables x1, …, xn, a homogeneous polynomial P satisfies P(t x1, …, t xn) = t^d P(x1, …, xn).

Examples help illustrate the concept. The monomial x1^a1 x2^a2 … xn^an is homogeneous of degree a1 + a2 + … + an. A polynomial like x^2 y is homogeneous of degree 3, while x^2 + y^2 is homogeneous of degree 2. A polynomial such as x^2 + x y + 1 is not homogeneous. The zero polynomial is typically regarded as homogeneous of every degree, though some conventions treat its degree as undefined.

The set of homogeneous polynomials of a fixed degree d in n variables forms a k-vector space. Its dimension is binomial(n + d − 1, d), and a standard basis consists of all monomials x1^{a1} x2^{a2} … xn^{an} with sum ai = d. Any polynomial P can be uniquely decomposed into a sum of homogeneous components: P = P0 + P1 + … + Pdeg(P), where each Pi is homogeneous of degree i.

Applications and importance. Homogeneous polynomials are central in projective geometry because their equations define projective varieties, which are invariant under scaling. They also appear in invariant theory, representation theory, and various areas of algebraic geometry and analysis, where the degree-d forms carry essential geometric and algebraic information about the objects they define.