arXiv subjects

Zhi-Lin Zhang

Publications and source records attributed to Zhi-Lin Zhang.

8 recordsLinked to original sources

arXiv2026

Vanishing-set inclusion and character degrees in finite $p$-groups

Let $G$ be a finite $p$-group and let $χ,ψ\in\operatorname{Irr}(G)$. We prove that $$\operatorname{Van}(χ)\subseteq\operatorname{Van}(ψ) \quad\Longrightarrow\quad χ(1)\midψ(1),$$ where $\operatorname{Van}(θ)=\{g\in G:θ(g)=0\}$. Consequently, two irreducible characters of a finite $p$-group with the same vanishing set have the same degree. This answers a question attributed to J. Sangroniz and recorded by Hung, Moretó, and Morotti, extending the previously known case in which one character has degree $p$. The same divisibility conclusion holds for irreducible characters of finite nilpotent groups.

math.GR↗

arXiv2026

Solubility from a disconnected common-divisor graph on $p$-regular conjugacy-class sizes

Let $G$ be a finite group and let $p$ be a prime. We prove that if the common-divisor graph on the nontrivial conjugacy-class sizes of $p$-regular elements of $G$ is disconnected, then $G$ is soluble. This settles the remaining case identified by Camina and collaborators. Their structure results provide an abelian normal $p'$-subgroup $M$. We choose a noncentral $r$-element $x\in M$, with $r\ne p$, such that $[G:C_G(x)]$ is coprime to $pr$, and show that $C_G(x)/M$ is a $\{p,r\}$-group. Thus $C_G(x)/M$ is a Hall $\{p,r\}$-subgroup of $G/M$. The separation between the two graph components then implies that every $p$-regular conjugacy-class size in $G/M$ has all its prime divisors in $\{p,r\}$. A theorem of Dolfi and Lucido, together with Burnside's $p^a q^b$-theorem, yields the solubility of $G/M$ and hence of $G$.

math.GR↗

arXiv2026

Integral trace forms of cyclic number fields without tameness

Let $K/\mathbf{Q}$ be a cyclic number field and let $T_K=\bigl(\mathcal{O}_K,(x,y)\mapsto \operatorname{Tr}_{K/\mathbf{Q}}(xy)\bigr)$ be its integral trace lattice. We prove that two cyclic number fields have isometric integral trace lattices if and only if they have the same degree, signature, and signed discriminant. More generally, an isomorphism between the Dirichlet character groups of two finite abelian extensions that preserves the conductor and parity of every character yields an isometry of their integral trace lattices, equivariant for the dual identification of Galois groups. For cyclic fields, the degree, signature, and discriminant determine the character group together with these two functions. This proves Bolaños's conjecture for wildly ramified cyclic fields and gives the corresponding extension of the tame classification: in general, the signature must be retained as an additional invariant.

math.NT↗

arXiv2026

Group-ring methods for Alexander quandle rings

For an Alexander quandle on an abelian group $G$ with automorphism $ϕ$, we identify its augmentation powers and nonunital commutator subalgebra with explicit ideals in the ordinary group ring over any commutative unital coefficient ring. Each left-normed augmentation power is also the span of all products with the same number of factors, independently of parenthesization. Over $\mathbb{Z}$, we determine every successive augmentation quotient for Alexander quandles on nontrivial finite cyclic groups. For an even dihedral quandle of order $m \ge 4$, every quotient in degree $r \ge 2$ is $(\mathbb{Z}/(m/2)\mathbb{Z})^2$; this proves the proposed replacement for the disproved order-$m$ prediction. For the integral quandle ring of $\operatorname{Core}(\mathbb{Z})$, we describe all augmentation powers, determine the nonunital commutator subalgebra, and prove that the difference of the basis elements indexed by 1 and 0 does not lie in the square of the augmentation ideal. When $(1-ϕ)G$ is a finite $p$-group, the integral filtration is separated, so every nonzero integral idempotent has augmentation one. Fourier analysis shows that finite odd-order commutative Alexander quandles have only zero and the basis elements as integral idempotents. Combined with the midpoint description of medial commutative quandles, this gives the same classification for every finite medial commutative quandle. Finally, the complex quandle algebra of the core of any nontrivial finite abelian group of odd order has right Peirce spectrum $\{0,1,-1\}$, proving the conjectured odd-order dihedral spectrum.

math.RA↗

arXiv2026

A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups

For positive integers $q$ and $m$ with $m\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every $q\geq 1$, including an independent proof of the previously known case $q=1$. Let $P_{m,q}$ be the semigroup polynomial, and set $t=m-2q-3$, $L=2(q+1)(m+1)-1$, and $D=°P_{m,q}=2qm+2q+2$. For $m\geq 2q+4$, let $ρ_1,\ldots,ρ_D$ be the roots of $P_{m,q}$, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives $\left|\sum_{j=1}^{D}ρ_j^{-L}\right|=L\left|[x^L]\log P_{m,q}(x)\right|\geq Lt-1>D$. The strict inequality forces $P_{m,q}$ to have a root off the unit circle. Hence $S_{m,q}$ is noncyclotomic whenever $m\geq 2q+4$. The boundary case $m=2q+3$ has embedding dimension $3$ and is cyclotomic. Therefore $S_{m,q}$ is cyclotomic if and only if $m=2q+3$.

math.NT↗

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&λI_2&D\\ 0&0&λI_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ λ\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

Equality cases for matrix spaces with bounded-rank commutators

Let $0\leq k < n$, and let $\mathcal V\subseteq M_n(\mathbb C)$ be a complex linear subspace satisfying $\operatorname{rank}[S,T]\leq k$ for all $S,T\in\mathcal V$. Omladič, Radjavi, and Šivic proved the sharp bound $\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1$ and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, $\mathcal V$ consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximal-dimensional commuting subspace of $M_{n-k}(\mathbb C)$. For $n-k\geq4$, these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions $2$ and $3$, the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian. For $2\leq k\leq n-2$, we determine all of its irreducible components. If $n-k\geq4$, there are exactly two components when $n-k$ is even and exactly four when $n-k$ is odd; if $n-k\in\{2,3\}$, there are exactly two. When $n-k\geq4$, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When $n-k\in\{2,3\}$, the two components are obtained by varying the invariant $k$-dimensional subspace and the maximal-dimensional commuting subspace on the quotient, together with their transposes.

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 $K_n=\mathbb{Q}(ρ_n)$, where $ρ_n$ 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--Pethő--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↗