A group of mathematicians has proved that, for any fixed number of mutually commuting matrices, the size of the algebra they generate grows at most linearly with the matrix dimension. In precise terms, if $A_1, ld,A_m rianglelefteq extrm{M}_n(oldsymbol{F})$ commute pairwise, then the dimension of the algebra $oldsymbol{F}[A_1, ld,A_m]$ cannot exceed $(2^m-1)(n-1)+1$. This settles an open question raised by Bergman.

The proof uses a combinatorial argument about maximal rank and a co-length inequality in the polynomial ring $oldsymbol{F}[x_1, ld,x_m]$. The authors select a vector $v$ that gives the largest possible subspace $W=oldsymbol{F}[A_1, ld,A_m]v$, and then analyse how the annihilator ideals of $V$ and $W$ interact inside the free polynomial algebra. This leads to the linear upper bound above.

The authors also examine the sequence $c_m(oldsymbol{F})$, the worst‑case ratio of the algebra dimension to the matrix size, and show that $rac{ ext{log}igl(c_m(oldsymbol{F})igr)}{m} o ext{log}igl(2igr)$ as $m$ grows. In other words, the growth rate of $c_m(oldsymbol{F})$ is exponential with base $2$, and .

It is important to note that the result does not rely on any geometric or representation–theoretic simplifications that work in lower dimensions. For $miglreve{}igr eq2$, the variety of commuting $m$‑tuples no longer remains irreducible once $n ight extgreater=4$, so previous methods fail. The new proof avoids these obstacles and applies uniformly to all $m$ and $n$.

In summary, the paper provides a clean linear bound for the dimension of algebras generated by commuting matrices, clarifies their asymptotic growth, and demonstrates that earlier geometric approaches cannot be extended to general settings. The result is purely theoretical and does not directly affect algorithmic or practical applications involving matrices.