  Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n
\in {\mathbb Z}$ by putting \[ E{a}(\alpha,\beta):=\sum_{|n|\leq N}a(n)
e(\alpha n^3 + \beta n), \] we obtain the conjectured tenth moment estimate \[
\| E{a} \|_{L^{10}({\mathbb T}^2)}\lesssim_\epsilon N^{\frac{1}{10}+\epsilon}
\|a\|_{\ell^2({\mathbb Z})}. \] We obtain related conclusions when the curve
$(x,x^3)$ is replaced by $(\phi_1(x), \phi_2(x))$ for suitably independent
polynomials $\phi_1(x),\phi_2(x)$ having integer coefficients.
