[Paper Review] The probability of rectangular unimodular matrices over $\F_q[x]$

Xiangqian Guo, Guangyu Yang|arXiv (Cornell University)|Jul 18, 2011

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper determines the natural density of $k \times n$ unimodular matrices over the polynomial ring $\mathbb{F}_q[x]$, showing that the probability a random $k \times n$ matrix can be extended to an invertible $n \times n$ matrix is $\prod_{j=n-k+1}^{n} \zeta_q(j)^{-1}$, where $\zeta_q(j)$ is the $q$-zeta function. The result generalizes Dirichlet's density theorem on coprime integers to polynomial rings over finite fields using probabilistic density limits and Euler product structures.

ABSTRACT

In this note, we compute the probability that a $k imes n$ matrix can be extended to an $n imes n$ invertible matrix over $\F_q[x]$, which turns out to be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's density theorem on the co-prime integers and its various generalizations are also presented.

Motivation & Objective

• To extend the matrix form of Dirichlet’s density theorem from integer matrices to matrices over polynomial rings $\mathbb{F}_q[x]$.
• To define and compute the natural density of unimodular $k \times n$ matrices over $\mathbb{F}_q[x]$ for $k \leq n$.
• To establish a connection between the probability of unimodularity and the $q$-zeta function, generalizing the classical coprime probability over $\mathbb{Z}$.
• To provide a rigorous probabilistic framework for matrix unimodularity in polynomial rings using asymptotic density over increasing degree bounds.

Proposed method

• Define a natural density on $\mathcal{M}_N$, the set of $k \times n$ matrices with entries in polynomials of degree at most $N$, via the limit $D(S) = \lim_{N \to \infty} |S \cap \mathcal{M}_N| / |\mathcal{M}_N|$.
• Use the $q$-zeta function $\zeta_q(j) = \prod_{f \text{ irred}} (1 - q^{-j \deg f})^{-1}$ to model the probability of coprimality among minors.
• Apply inclusion-exclusion and Euler product techniques to compute the density of matrices whose full-rank minors are coprime.
• Establish bounds on the density of matrices divisible by a fixed irreducible polynomial $f$ using $D(H_f) \leq 2 / q^{2 \deg f}$.
• Use the convergence of the product $\prod_{m=1}^\infty (1 - q^{-jm})^{\varphi_m}$ to show the limit exists and equals $\zeta_q(j)^{-1}$.
• Prove the main result by taking the limit over increasing sets of irreducible polynomials and using the convergence of the product over all irreducibles.

Experimental results

Research questions

• RQ1 What is the natural density of $k \times n$ unimodular matrices over $\mathbb{F}_q[x]$ for $k \leq n$?
• RQ2 How does the probability of unimodularity in $\mathbb{F}_q[x]$ relate to the $q$-zeta function and Euler products?
• RQ3 Can Dirichlet’s classical result on coprime integers be generalized to matrices over polynomial rings using a natural density framework?
• RQ4 What is the asymptotic probability that a random $k \times n$ matrix over $\mathbb{F}_q[x]$ can be extended to an invertible $n \times n$ matrix?

Key findings

• The natural density of $k \times n$ unimodular matrices over $\mathbb{F}_q[x]$ is $\prod_{j=n-k+1}^{n} \zeta_q(j)^{-1}$, where $\zeta_q(j)$ is the $q$-zeta function.
• This result generalizes the classical probability $\zeta(n)^{-1}$ that $n$ random integers are coprime to the matrix case over $\mathbb{F}_q[x]$.
• The probability that a $1 \times n$ matrix over $\mathbb{F}_q[x]$ is unimodular is $\zeta_q(n)^{-1}$, matching the $q$-analogue of Dirichlet’s theorem.
• The proof relies on bounding the density of matrices divisible by irreducible polynomials and using the convergence of Euler products over all monic irreducibles.
• The limit $\lim_{N \to \infty} |E \cap \mathcal{M}_N| / |\mathcal{M}_N|$ exists and equals the infinite product $\prod_{j=n-k+1}^{n} \zeta_q(j)^{-1}$.
• The method establishes that the density of unimodular matrices is determined by the product of local densities at each irreducible polynomial, analogous to the Chinese Remainder Theorem in number theory.