 Homogeneous polynomial

A nonzero polynomial is homogeneous of degree $d$ when every monomial with nonzero coefficient has total polynomial degree $d$. Equivalently, the formal polynomial identityholds with an extra indeterminate $T$. For example, $X^2+3XY+Y^2$ is homogeneous of degree two, whereas $X^2+Y$ mixes degrees. The zero polynomial is commonly included in every homogeneous component.

Over an infinite field, testing the scaling equation at all scalar values also detects homogeneity. Over a finite field, equality of polynomial functions is weaker than a formal polynomial identity: $X^q-X$ vanishes everywhere on ${\mathbb{F}}_q$ despite mixing degrees. The monomial definition avoids that ambiguity. Homogeneity is what lets a zero set descend to projective space, since scaling a nonzero coordinate representative does not change whether the polynomial vanishes.

A **homogeneous polynomial** is a polynomial whose terms all have the same total degree. In more formal terms, a polynomial \( P(x_1, x_2, \ldots, x_n) \) is called homogeneous of degree \( d \) if every term in the polynomial is of degree \( d \).