论文标题
多项式进程在有限领域的真正复杂性
True complexity of polynomial progressions in finite fields
论文作者
论文摘要
有限场中多项式进程的真实复杂性对应于最小的高级范围规范，该规范控制着大型特征的有限场上进展的计数算子。我们给出了一个猜想，将真正的复杂性与进度术语之间的代数关系联系起来，我们证明了这一点用于许多进度，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，我们证明了在有限场的子集中某些复杂性1的计数的渐近学。在此过程中，我们获得了某些多项式进行的等分分配结果，类似于Green和Tao证明的线性形式系统的计数引理。
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.
