On the smallest number of generators and the probability of generating an algebra
Let \(R\) be an order in \(K\), an extension of degree \(d\) of the rationals, and let \(\omega_1,\dots,\omega_d\) be an integral basis of \(R\). For positive \(N\) let \(R(N)=\{\sum_{i=1}^dx_j\omega_j:\;x_j\in Z,\;|x_j|\leq N\}\). The authors show that if \(f_1,\dots,f_s\) are polynomials in \(n\) variables over \(R\), and \(S\) denotes the set of \({\mathbf x}\in R^n\) such that the set \(\{f_i({\mathbf x}):\;i=1,2,\dots,s\}\) generates the unit ideal, then NEWLINE\[NEWLINE\lim_{N\to\infty}{\#(S\cap R(N))\over (2N+1)^{dn}}=\prod_M\left(1-{t_M\over N(M)^n}\right),NEWLINE\]NEWLINE where \(M\) runs over all maximal ideals of \(R\), and \(t_M\) is the number of common zeros of \(f_i \bmod M\).NEWLINENEWLINEThis result is used to establish the following assertion on sets generating an \(R\)-algebra. Let \(A\) be an \(R\)-algebra which is free of finite rank \(m\) as an \(R\)-module. If \(e_1,\dots,e_m\) is an \(R\)-basis of \(A\), then the set \(\{e_i\omega_j\}\) forms a \(Z\)-basis of \(A\). For positive integers \(N\) let \(A(N)\) be the set of elements of \(A\) whose all coefficients in that basis lie in the interval \([-N,N]\). Moreover let \(Gen_k(A)\) be the set of all \(k\)-element generators of \(A\) as an \(R\)-algebra. The authors define the density of the family of these sets by NEWLINE\[NEWLINE\text{den}_k(A)=\lim_{N\to\infty}{(\# Gen_k(A)\cap A(N)^k)\over\#A(N)},NEWLINE\]NEWLINE and show that this limit exists and can be expressed as the product of the analogously defined local densities.NEWLINENEWLINEThis is used to answer a question of Kapovich, asking for the probability that \(k\) elements chosen at random generate the group \(Z^n\). This probability turned out to be equal to the product NEWLINE\[NEWLINE\prod_{m=k-n+1}^k{1\over\zeta(m)},NEWLINE\]NEWLINE \(\zeta(s)\) being Riemann's zeta-function. The authors provide also several other applications, dealing with the smallest numbers of generators of \(R\)-algebras, generating sets of matrix algebras over finite fields and of generating sets of finite products of matrix algebras over rings of algebraic integers.
scientific article; zbMATH DE number 6064710