Rota's basis conjecture holds for random bases of vector spaces

TL;DR

This paper proves that Rota's basis conjecture is almost surely true for large random bases of vector spaces over finite fields and subsets of finite sets, confirming the conjecture in a probabilistic setting.

Contribution

It demonstrates that Rota's basis conjecture holds with high probability for large random bases over finite fields and finite subsets, extending the understanding of the conjecture's validity.

Findings

Rota's basis conjecture holds with probability approaching 1 as dimension increases for random bases over finite fields.

The conjecture is also valid with high probability for bases chosen uniformly from subsets of finite sets.

Results apply to both finite fields and finite subsets, broadening the conjecture's probabilistic validation.

Abstract

In 1989, Rota conjectured that, given $n$ bases $B_1,\dots,B_n$ of the vector space $\mathbb{F}^n$ over some field $\mathbb{F}$, one can always decompose the multi-set $B_1\cup \dots \cup B_n$ into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases $B_1,\dots,B_n$. More specifically, our first result shows that Rota's basis conjecture holds with probability $1-o(1)$ as $n\to \infty$ if the bases $B_1,\dots,B_n$ are chosen independently uniformly at random among all bases of $\mathbb{F}_q^n$ for some finite field $\mathbb{F}_q$ (the analogous result is trivially true for an infinite field $\mathbb{F}$). In other words, the conjecture is true for almost all choices of bases $B_1,\dots,B_n\subseteq \mathbb{F}_q^n$. Our second, more general, result concerns random bases $B_1,\dots,B_n\subseteq S^n$ for some given finite subset $S\subseteq \mathbb{F}$ (in other words, bases $B_1,\dots,B_n$ where all vectors have entries in $S$). We show that when choosing bases $B_1,\dots,B_n\subseteq S^n$ independently uniformly at random among all bases that are subsets of $S^n$, then again Rota's basis conjecture holds with probability $1-o(1)$ as $n\to \infty$.