True complexity of polynomial progressions in finite fields

The true complexity of a polynomial progression in finite fields corresponds to the smallest-degree Gowers norm that controls counting operator over large characteristic. We give conjecture relates algebraic relations between terms progression, and we prove it for number progressions, including $x,\; x+y,\; x+y^2,\; x+y+y^2$ x+2y,\; x+y^2$. As corollary, an asymptotic count certain progressions 1 subsets fields. In process, obtain equidistribution result analogous lemma systems linear forms proved by Green Tao.