关于生成元的最小个数与生成代数的概率
On the smallest number of generators and the probability of generating an algebra
摘要：
本文研究数域上阶代数生成元集的代数性质和渐近性质。设$A$是代数数域中阶$R$上的结合代数.我们假设$A$是有限秩的自由$R$ -模.我们开发了一种技术来计算生成器的最小数量的$A$。例如，我们证明了环$M_3(\mathbb{Z})^{k}$允许两个生成元当且仅当$k\leq 768$。对于一个给定的正整数$m$，我们定义生成它的$A$的所有有序$m$ -元组的集合的密度为一个$R$ -代数.我们将这个密度表示为$R$的极大理想上的某个无穷乘积，并从概率的角度解释所得公式。例如，我们证明了2个随机$3\times 3$矩阵生成环$M_3(\mathbb{Z})$的概率等于$(ζ(2)^2 ζ(3))^{-1}$，其中$ζ$是黎曼zeta函数。
英文摘要：
In this paper we study algebraic and asymptotic properties of generating sets of algebras over orders in number fields. Let $A$ be an associative algebra over an order $R$ in an algebraic number field. We assume that $A$ is a free $R$-module of finite rank. We develop a technique to compute the smallest number of generators of $A$. For example, we prove that the ring $M_3(\mathbb{Z})^{k}$ admits two generators if and only if $k\leq 768$. For a given positive integer $m$, we define the density of the set of all ordered $m$-tuples of elements of $A$ which generate it as an $R$-algebra. We express this density as a certain infinite product over the maximal ideals of $R$, and we interpret the resulting formula probabilistically. For example, we show that the probability that 2 random $3\times 3$ matrices generate the ring $M_3(\mathbb{Z})$ is equal to $(ζ(2)^2 ζ(3))^{-1}$, where $ζ$ is the Riemann zeta-function.
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