Prof. Tao's answer is excellent. I also found two research papers answering the question so I list them below as complementary reference:

1. G.I.Arkhipov and K.I.Oskolkov, On a special trigonometric series and its applications, 1989 Math. USSR Sb. 62 145 Link to the article: http://iopscience.iop.org/0025-5734/62/1/A10 See Theorem 1.

2. E.S.Stein and S.Wainger, Discrete analogues of singular Radon transform, Bulletin of AMS 1990 Link to the article: http://www.ams.org/journals/bull/1990-23-02/S0273-0979-1990-15973-7/S0273-0979-1990-15973-7.pdf See the Lemma in Section 6

The key of their results is the upper bound depend only on the degree (not the coefficients, i.e. $x$ and $y$ in the question) of the polynomial.

Prof. Tao's answer is excellent. I also found two research papers answering the question so I list them below as complementary reference:

1. G.I.Arkhipov and K.I.Oskolkov, On a special trigonometric series and its applications, 1989 Math. USSR Sb. 62 145 Link to the article: http://iopscience.iop.org/0025-5734/62/1/A10 See Theorem 1.

2. E.S.Stein and S.Wainger, Discrete analogues of singular Radon transform, Bulletin of AMS 1990 Link to the article: http://www.ams.org/journals/bull/1990-23-02/S0273-0979-1990-15973-7/S0273-0979-1990-15973-7.pdf See the Lemma in Section 6

The key of their results is that the upper bound depends only on the degree (not the coefficients, i.e. $x$ and $y$ in the question) of the polynomial.