Limit distribution of the number of solutions of a system of stochastic linear homogeneous equations with a uniform coefficient matrix over a finite local ring
Let \(R\) be a finite local ring with radical \(I(R)\). Consider the system of linear equations (SLE): \[ (1)\quad Ax^\downarrow=0^\downarrow,\qquad (2)\quad By^\downarrow=0^\downarrow, \] where \(A\) and \(B\) are \((n+s)\times n\) matrices over \(R\) and \(s\) is an integer constant. Assume that the elements of each matrix \(A\) and \(B\) are jointly independent random variables uniformly distributed on the set \(R\) and \(I(R)\), respectively. Denote by \(\nu_{n,s}\) the number of solutions of SLE(1) and by \(\mu_{n,s}\) the number of solutions of SLE(2). Let \(s_0=-s\vee 0\), \(\pi_{n,s}(k)=P(\nu_{n,s}=q^k)\), \(\rho_{n,s}(k)= P(\mu_{n,s}(k)=q^k)\), \(k=0,1,\dots\). The exact expressions and the limit expressions (as \(n\to\infty)\) for the probabilities \(\pi_{n,s}(k)\), \(k=0,1, \dots\) and the limit (as \(n\to\infty)\) for \(\pi_{n,s}(-sN)\), where \(s\leq 0\) are obtained. The author proves a theorem which generalizes previous results.
system of stochastic linear homogeneous equations
scientific article; zbMATH DE number 1283637