Discrete restriction for $(x,x^3)$ and related topics

Abstract: 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.