扩展黎曼假设（ERH）下的确定性多项式分解
Deterministic polynomial factoring under the assumption of the Extended Riemann Hypothesis (ERH)
摘要：
在扩展黎曼假设（ERH）的假设下，考虑有限域上一元多项式的确定性分解问题。这项工作建立在Gao在$2001$中首次探索的方法基础上。一般的方法是隐式地构造一个以根为顶点、以多项式时间的可计算关系为边的图。如果该关联图证明是\emph{regular}，则该算法无法分解多项式。在我们的工作的第一部分中，我们加强边关系，使我们得到的图的结果集是高的图的子图，所有的都必须是\emph{regular}。在第二部分中，我们加强了这些图的正则性条件。这是通过找到一个并行的算法和$1$维Weisfeiler-Leman算法解决图同构问题。我们观察到，他们的算法背后的一般原则是通过计算该图的轨道的$1$维Weisfeiler-Leman近似来分离根。这就引出了一个自然的问题：这种近似是否可以改进。然后，我们继续展示如何隐式地计算这些图的轨道的$2$维Weisfeiler-Leman近似。我们的算法不能分解的多项式形成了\emph{strongly regular}图，它们的邻接矩阵集形成了一个称为\emph{Association scheme}的组合结构。
英文摘要：
We consider the problem of deterministically factoring a univariate polynomial over a finite field under the assumption of the Extended Riemann Hypothesis (ERH). This work builds upon the line of approach first explored by Gao in $2001$. The general approach has been to implicitly construct a graph with the roots as vertices and the edges formed by some polynomial time computable relation. The algorithm then fails to factor a polynomial if this associated graph turns out to be \emph{regular}. In the first part of our work we strengthen the edge relation so that the resulting set of graphs we obtain are subgraphs of Gao's graphs, all of which must be \emph{regular}. In the second part of our work we strengthen the regularity condition of these graphs. This is accomplished by finding a parallel between their algorithms and the $1$-dimensional Weisfeiler-Leman algorithm for solving the Graph Isomorphism problem. We observe that the general principle behind their algorithms is to separate the roots by computing the $1$-dimensional Weisfeiler-Leman approximation to the orbits of this graph. This leads us to the natural question of whether this approximation may be improved. We then go on to show how to implicitly compute the $2$-dimensional Weisfeiler-Leman approximation of the orbits of these graphs. The polynomials that our algorithm fails to factor form graphs that are \emph{strongly regular} and their set of adjacency matrices forms a combinatorial structure called an \emph{Association scheme}.
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