Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.06800

On the asymptotic Makar-Limanov rank conjecture

SiZhuo Yan·Hao Shen·Jianting Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $\Bbbk$ be an algebraically closed field of characteristic zero and let $f$ be a nonconstant polynomial in finitely many freely noncommuting variables. We prove the asymptotic Makar-Limanov rank conjecture, namely that the normalized rank of a value of $f$ can be made arbitrarily small by choosing matrices over $\Bbbk$ of finite size.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

SiZhuo Yan, Hao Shen, Jianting Yang. 2026-10-06. On the asymptotic Makar-Limanov rank conjecture. https://arxiv.org/abs/2610.06800

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Nilpotent Product Probability of Finite Rings

For a finite ring $R$, we investigate the probability that the product of two randomly chosen elements in $R$ is nilpotent, which we call the nilpotent product probability (NPP) of $R$ and denote by $P_{nil}(R)$. We derive bounds for $P_{nil}(R)$ that depend on the structure of $R$, and prove that among all commutative rings with identity, the class of local rings with residue field $\mathbb{Z}_{2}$ attains the maximum value of NPP, which is equal to $3/4$. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of $R$ and NPP of some subrings of the matrix ring over $R$.

math.RA↗
arXiv2026

Diagonalizable modular derivations on bigraded Poisson spaces and Koszul duality

For two bigraded spaces $k^{m||n}$ and $k^{n\wedge m}$ equipped with appropriate quadratic Poisson structures, Koszul duality identifies the mixed Poisson differential calculi and, in the unimodular case, their Poisson cohomologies as Batalin--Vilkovisky algebras. In the current paper we prove that Koszul duality induces an isomorphism of Batalin--Vilkovisky algebras between their Poisson cohomologies when the modular derivation of one algebra is diagonalizable, including non-unimodular cases.

math.RA↗
arXiv2026

Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type

We determine the first extension groups between simple restricted modules for the restricted simple Lie algebras of Cartan type over an algebraically closed field of characteristic $p\ge5$, which answers the restricted case of a problem of Benkart and Feldvoss. Extensions from parabolically induced modules are computed by finite restricted-cohomology complexes of the grading parabolic, and for the exceptional modules we obtain closed formulas in all four families. Neither the trivial module nor any nontrivial exceptional simple has self-extensions. In characteristic seven we determine the quiver of $S(3;1)^{(1)}$, which has 49 vertices and 566 arrows.

math.RA↗