论文标题
随机rédei矩阵的有效收敛
Effective convergence of coranks of random Rédei matrices
论文作者
论文摘要
我们给出了有效的估计，对$ r \ times r $rédei矩阵与Cohen-lenstra启发式方法预测的corank分布之间的$ l^1 $延伸。为此，我们指出了一类随机流程，我们称之为$ c $ - 传输。这些随机过程通过马尔可夫过程很好地近似，我们为此类过程提供了有效的千古定理。使用此工具，我们使有效的gerth \ cite {gerth}定理，以$ p = 2 $进行了对Cohen--lenstra启发式的研究。 格斯（Gerth）的作品引发了一系列发展，这些发展最近在史密斯（Smith \ Cite）的突破中达到了{史密斯}的突破。目前的工作将用于作者即将进行的工作，以实现史密斯想法的进一步应用，以实现二次领域的算术。为此，我们将主要结果扩展到了其他几个矩阵空间家族，这些矩阵空间在方程式上的积分点$ x^2 -dy^2 = l $ at $ d $都有不同。
We give effective estimates for the $l^1$-distance between the corank distribution of $r \times r$ Rédei matrices and the measure predicted by the Cohen--Lenstra heuristics. To this end we pinpoint a class of stochastic processes, which we call $c$-transitioning. These stochastic processes are well approximated by Markov processes, and we give an effective ergodic theorem for such processes. With this tool we make effective a theorem of Gerth \cite{Gerth} that initiated the study of the Cohen--Lenstra heuristics for $p = 2$. Gerth's work triggered a series of developments that has recently culminated in the breakthrough of Smith \cite{Smith}. The present work will be used in upcoming work of the authors on further applications of Smith's ideas to the arithmetic of quadratic fields. To this end we extend our main result to several other families of matrix spaces that occur in the study of integral points on the equation $x^2 - dy^2 = l$ as $d$ varies.
