Fourier methods and non degenerate polynomials

We use Fourier methods to give a shorter proof of Vu's result and extend it to a more general setting, namely: For any non-degenerate polynomials f(x,y)∈Fq[x,y] and A,B,C,D⊂Fq, either we have|A+C||B+D||f(A,B)|≳|A||B|q, or|A+C||B+D||f(A,B)|2≳|A|2|B|2|C||D|q.