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Seminarium i matematik

On the probability distribution of the number of solutions to linear congruence systems

Välkommen till föreläsningen i seminarieserien i matematik.

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 following problem:

  • How many $n\times m$-matrices over $\mathbb{Z}_{p^s}$ are there given that we have exactly $p^j$ solutions?

                         

For the case $s=1$ ($\mathbb{Z}_{p^s}$ is a field) George von Landsberg proved a general formula in 1893. However, there seems to be no general results for the case $s>1$ except when we have a unique solution ($j=0$). In this talk we present a general formula for the case when $j<s$ and $n\geq m$. We will use a generalization of Euler's $\varphi$-function and Gaussian binomial coefficients to express our formula.