On the polynomial szemerédi theorem in finite fields

Let P ₁ ; . . . ; P _m ∈ Z[y] be any linearly independent polynomials with zero constant term. We show that there exists γ > 0 such that any subset of F _q of size at least q ^1-γ contains a nontrivial polynomial progression x,x+P ₁ (y); . . . . x+P _m (y) provided that the characteristic of F _q is large enough.