In a DVR, why does $u=f(t,u)$, with $f$ a homogeneous polynomial of degree $3$ and $t$ a uniformizer, imply $\nu(u)=3$?

Question

In this answer by Georges Elencwajg, it is stated that

$$u=t^3-\dots-e_1e_2e_3u^3=\text{a homogeneous polynomial of degree $3$ in t,u}\quad(\ast)$$ [...]

Now in the local ring $\mathcal O_{C,P}$, a discrete valuation ring, equation $(\ast)$ implies that $u$ has valuation $3$.

In that context, $C$ is an algebraic curve and $t$ is a uniformizer for $\mathcal O_{C,P}$.

Why does this implication hold?