In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have the same total degree. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): x^5 + 2 x^3 y^2 + 9 x y^4

of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial Failed to parse (Missing texvc executable; please see math/README to configure.): x^3 + 3 x^2 y + z^7

A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A homogeneous polynomial of degree 1 is a linear form,. A homogeneous polynomial of degree 2 is a quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.

Algebraic form, or simply form, is another term for homogeneous polynomial. These then generalise from quadratic forms to degrees 3 and more, and have in the past also been known as quantics (a term that originated with Cayley). To specify a type of form, one has to give its degree of a form, and number of variables n. A form is over some given field K, if it maps from Kⁿ to K, where n is the number of variables of the form.