Homogeneous polynomial

A polynomial whose nonzero monomials all have the same total degree. Scaling every variable by a factor scales its value by that factor raised to the polynomial’s degree.

Polynomial: An expression formed from variables and coefficients using addition, subtraction, and nonnegative integer powers. Homogeneous polynomials are a special kind of polynomial.
Scaling: A transformation that multiplies quantities by a common factor. Homogeneity describes exactly how the polynomial changes under scaling of all variables.
Homogeneous coordinates: Coordinates that represent a projective point by nonzero tuples identified up to common scaling. Homogeneous polynomials provide equations that remain well-defined under coordinate rescaling.
Nonhomogeneous polynomial: A polynomial containing nonzero monomials of different total degrees. It violates the defining equal-degree condition and generally lacks the same scaling law.
Monomial: A product of a coefficient and variables raised to nonnegative integer powers. Each term is a monomial, and its degree contributes to homogeneity.
Euler's homogeneous function theorem: A theorem relating a differentiable homogeneous function to its first partial derivatives. It converts the scaling property into an identity involving the polynomial and its derivatives.
Algebraic variety: A geometric set defined as the common zeros of polynomial equations. Homogeneous equations define projective varieties and affine cones.
Multihomogeneous polynomial: A polynomial homogeneous separately in multiple groups of variables, with a degree assigned to each group. It refines ordinary homogeneity when variables belong to distinct groups.
Total degree: The greatest sum of variable exponents among a polynomial’s monomials. Homogeneity requires every nonzero monomial to have the same total degree.
Projective geometry: The study of geometric properties invariant under projection and perspective transformations. Homogeneous equations define zero sets that are unchanged when coordinates are scaled together.

Linked from 8 pages