We obtain a necessary and sufficient condition on a polynomial $P(t_1,t_2)$ for $L^2$ boundedness of the discrete double Hilbert transforms along polynomial surfaces $(t_1,t_2,P(t_1,t_2))$ for $1 < p < infty$ and $L^{p}$ boundedness of the operator with the sufficient condition.  This is joint work with Joonil Kim.