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.