Counting linear congruence systems with a fixed number of solutions

TL;DR

This paper develops recursive formulas and explicit methods for counting matrices over rings of integers modulo prime powers with a fixed number of solutions, extending classical results to more complex algebraic structures.

Contribution

It introduces recursive techniques and explicit formulas for counting matrices with a specified number of solutions over rac{p^s}{ ext{ring of integers modulo } p^s}, generalizing known results for prime fields.

Findings

Derived recursive methods for counting matrices with fixed solution counts.

Provided explicit formulas for cases where the number of solutions is at most p^s.

Calculated probabilities related to gcd of determinants and solution counts.

Abstract

For a prime $p$ and a positive integer $s$ consider a homogeneous linear system over the ring $\mathbb{Z}_{p^s}$ (the ring of integers modulo $p^s$) described by an $n \times m$-matrix. The possible number of solutions to such a system is $p^j$, where $j=0,1,\ldots, sm$. We study the problem of how many $n \times m$-matrices over $\mathbb{Z}_{p^s}$ there are given that we have exactly $p^j$ homogeneous solutions. For the case $s=1$ (when $\mathbb{Z}_{p^s}$ is a field) George von Landsberg proved a general formula in 1893. However, there seems to be few published general results for the case $s>1$ except when we have a unique solution ($j=0$). In this article we present recursive methods for counting such matrices and present explicit formulas for the case when $j\le s$ and $n\ge m$. We will use a generalization of Euler's $\phi$-function and Gaussian binomial coefficients to express our formulas. As an application we compute the probability that gcd$(\det(A),p^s)$ gives the number of solutions to the quadratic system $Ax=0$ in $\mathbb{Z}_{p^s}$.