  Let $\operatorname{r_{max}}(n,d)$ be the maximum Waring rank for the set of
all homogeneous polynomials of degree $d>0$ in $n$ indeterminates with
coefficients in an algebraically closed field of characteristic zero. To our
knowledge, when $n,d\ge 3$, the value of $\operatorname{r_{max}}(n,d)$ is known
only for $(n,d)=(3,3),(3,4),(3,5),(4,3)$. We prove that
$\operatorname{r_{max}}(3,d)=d^2/4+O(d)$ as a consequence of the upper bound
$\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$.
