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.

中文翻译：

---

有限域中多项式级数的真实复杂性

有限域中多项式级数的真正复杂性对应于控制大特征有限域级数的计数算子的最小次高尔斯范数。我们给出了一个猜想，将真正的复杂性与级数项之间的代数关系联系起来，并为许多级数证明了它，包括$x, x+y, x+y^{2}, x+y+y^{2}$和$x, x+y, x+2y, x+y^{2}$. 作为推论，我们证明了有限域子集中某些复杂度为 1 级数的计数的渐近性。在这个过程中，我们得到了某些多项式级数的等分布结果，类似于 Green 和 Tao 证明的线性系统的计数引理。