Homogeneous polynomial

Summary

A homogeneous polynomial is a polynomial in several variables whose nonzero terms all have the same total degree. Such polynomials are also called forms; the quadratic forms are the homogeneous polynomials of degree $2$. A homogeneous polynomial of degree $d$ satisfies $f(\lambda x) = \lambda^d f(x)$, so its zero set is a union of lines through the origin. This is why homogeneous polynomials describe the subsets of projective space, and why they appear in the definition of quasi-algebraically closed fields.