Zhi-Lin Zhang | Searcharxiv

arXiv2026

The $6\times6$ equality case of matrix spaces with rank-two commutators

Let $\mathcal V\subseteq M_6(\mathbb C)$ be a $17$-dimensional linear subspace such that $ \operatorname{rank}[S,T]\leq2 \quad(S,T\in\mathcal V). $ We prove that $\mathcal V$, or its transpose, is conjugate to the algebra $ \left\{ \begin{pmatrix} A&B&C\\ 0&\lambda I_2&D\\ 0&0&\lambda I_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ \lambda\in\mathbb C \right\}. $ Consequently, the corresponding closed algebraic locus in $\operatorname{Gr}(17,M_6(\mathbb C))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\operatorname{Fl}(2,4;6)$. We also prove that the Zariski tangent space at $\mathcal A$ of the corresponding closed algebraic locus is equal to the tangent space to the conjugacy orbit of $\mathcal A$.

math.RA↗

arXiv2026

Equalities among simplest quartic fields: a complete classification

For a positive integer $n$, let $f_n(X)=X^4-nX^3-6X^2+nX+1$ and let $K_n=\mathbb{Q}(\rho_n)$, where $\rho_n$ is a root of $f_n$. We determine all coincidences among these fields: for distinct positive integers $m,n$, $K_m=K_n \Longleftrightarrow \{m,n\}\in\{\{1,103\},\{2,22\},\{4,956\}\}$. Thus the three previously known equalities are the only ones. This extends Hoshi's finite-range classification to all positive integral parameters and, in particular, subsumes the uniqueness results of Pincus and Washington. The proof combines Hoshi's correspondence between equal simplest quartic fields and primitive solutions of a quartic Thue equation with estimates of Lettl--Peth\H{o}--Voutier for rational approximations to two of its real roots. A Gaussian-integer identity yields a lower bound for the denominator of the resulting rational approximation; the continued-fraction information and the approximation estimates then exclude every parameter exceeding $1000$.

math.NT↗