Homogeneous polynomial

Remarks

Example: $f = x^2 y + 3 y z^2 - z^3$ in $\Q[x, y, z]$ is homogeneous of degree $3$. Check: the monomials $x^2 y$, $y z^2$ and $z^3$ have total degrees $2 + 1$, $1 + 2$ and $3$.

Non-example: $g = x^2 + y$ is not homogeneous, because its monomials have total degrees $2$ and $1$.

The polynomial $f$ is homogeneous of degree $d$ if and only if $f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$ in $R[\lambda, x_1, \ldots, x_n]$. Every polynomial is in a unique way a finite sum $f = f_0 + f_1 + \cdots$ where each $f_d$ is homogeneous of degree $d$; these are the homogeneous components of $f$. A homogeneous polynomial is also called a form; forms of degree $1$, $2$ and $3$ are linear, quadratic and cubic forms. In one variable, the nonzero homogeneous polynomials of degree $d$ are the $c x^d$ with $c \ne 0$, whose degree is $d$.