论文标题
整数上的矩阵的多项式方程模拟素数和随机矩阵的凝聚力
Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix
论文作者
论文摘要
给定一个prime $ p $和一个正整数$ k $，让$ \ mathrm {m} _ {n}（\ Mathbb {z}/p^{k} \ Mathbb {z}）$是$ \ \ \ \ m mathbb {z} z}/p^abb的$ n \ times n $矩阵的环。我们考虑解决方案的数量$ x \ in \ mathrm {m} _ {n}（\ mathbb {z}/p^{k} \ mathbb {z}）$ to polyenmial方程$ p（x）= 0 $ $（\ mathbb {z}/p^{k} \ mathbb {z}）[t] $，其还原模型$ p $在有限字段$ \ mathbb {f} _ {p} $ p $元素的$上是无方形的。注意$ p（x）= 0 $，仅当$ \ mathrm {cok}（p（x））\ simeq（\ mathbb {z}/p^{k} \ mathbb {z}）^{n} $ $ \ mathrm {cok}（p（x））$ p（x）$ of ISOMORPHISMS，其中$ x $是$ \ Mathrm {M {M} _ {n}（\ MathBb {Z}/p^{k}/p^{k} {k} \ mathbb {Z} Z}）中的均匀随机矩阵。当我们修复$ x $ modulo $ p $的残基类时，此分布涉及明确的公式。当$ \ mathbb {f} _ {p} [t] $ modulo $ p $不可记述时，我们证明了特殊情况下的猜想。我们说明我们获得的分布与Cohen-Lenstra分布密切相关。我们的证明涉及在$ \ mathbb {z}/p^{k} \ mathbb {z} $的$ \ mathbb {z}/p^{z} $上的代数和组合参数，并在先前的Cheong和Kaplan作品上建立。
Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation $P(X) = 0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$ if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, we give a conjectural generalization of counting solutions to $P(X) = 0$ as the distribution of the cokernel $\mathrm{cok}(P(X))$ of $P(X)$ up to isomorphisms, where $X$ is a uniform random matrix in $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$. This distribution involves an explicit formula when we fix the residue class of $X$ modulo $p$. We prove this conjecture for the special case when the image of $P(t)$ in $\mathbb{F}_{p}[t]$ modulo $p$ is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution. Our proof involves algebraic and combinatorial arguments in linear algebra over $\mathbb{Z}/p^{k}\mathbb{Z}$ and builds upon a previous work of Cheong and Kaplan.
