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