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