  In this paper we prove that the complex polynomial Bohnenblust-Hille constant
for $2$-homogeneous polynomials in ${\mathbb C}^2$ is exactly
$\sqrt[4]{\frac{3}{2}}$. We also give the exact value of the real polynomial
Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb R}^2$.
Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille
constant for polynomials in ${\mathbb R}^2$ of higher degrees.
