homogeneous polynomial

A polynomial $P(x_1,\mathrm{\cdots },x_n)$ of degree $k$ is called homogeneous if$P(c{x}_1,\mathrm{\cdots },c{x}_n)=c^{k}P(x_1,\mathrm{\cdots },x_n)$ for all constants $c$.

An equivalent definition is that all terms of the polynomial have the same degree (i.e. $k$).

Observe that a polynomial $P$ is homogeneous iff $\mathrm{deg}P=\mathrm{ord}P$.

As an important example of homogeneous polynomials one can mention the symmetric polynomials.