A 2-dimensional Generalization Of SáRK?ZY's Theorem In Function Fields

Let Fq[t]be the polynomial ring over the finite field Fq of q elements.For N E N let GN be the set of all polynomials in Fq[t]of degree less than N.Suppose that the characteristic of Fq is greater than 2 and A(?)GN2.If(d,d2)(?)A-A={a-a':a,a'?A} for any d?Fq[t]\{0},we prove that |A|?Cq2N logN/N.where the constant C depends only on q.By using this estimate,we extend Sárk?zy's theorem in function fields to the case of a finite family of polynomials of degree less than 3.