arXiv subjects

Hanbin Zhang

Publications and source records attributed to Hanbin Zhang.

10 recordsLinked to original sources

arXiv2026

On zero-sum polytopes: reciprocity, rigidity, and cyclic sieving

Let $G$ be a finite abelian group of order $n$, and let $\mathsf M(G,m)$ denote the set of zero-sum sequences over $G$ of length $m$. We introduce the zero-sum polytope $\mathcal P_G$, a rational polytope of dimension $n-1$, whose lattice points encode zero-sum sequences: \[ |\mathsf M(G,m)|=|m\mathcal P_G\cap \mathbb Z^n|. \] This naturally realizes the enumeration of zero-sum sequences as a problem in rational Ehrhart theory, which leads to a combinatorial reciprocity theorem identifying the negative evaluations of the corresponding counting quasipolynomial with zero-sum sequences of full support. Our main results establish a face-stratified rigidity for zero-sum polytopes: whenever two such polytopes have equal total lattice point counts at specific dilations, the dimension-wise open-face strata are equinumerous. Moreover, we study the natural $\operatorname{Aut}(G)$-action on $\mathcal P_G$, derive equivariant generating functions and reciprocity formulas, and obtain cyclic sieving phenomena for natural cyclic actions.

math.CO↗

arXiv2026

On immanants of Cayley tables

Let $G$ be a finite abelian group of order $n$ and let $\mathcal M_G=(x_{a+b})_{a,b\in G}$ be the Cayley table of $G$. Let $\text{imm}_\lambda(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $\lambda$ and $\mathcal I_\lambda(G)$ be the number of formally different monomials occurring in $\text{imm}_\lambda(\mathcal M_G)$ (in particular, we denote by $\mathcal P(G)$ (resp. $\mathcal D(G)$) for the corresponding quantity for $\text{per}(\mathcal M_G)$ (resp. $\text{det}(\mathcal M_G)$) for simplicity). The study of $\mathcal P(G)$ and $\mathcal D(G)$ lies at the intersection of algebraic combinatorics and additive combinatorics. In this paper, we prove the following results. (1) If $|G|$ is a prime power, then $$\mathcal P(G)=\mathcal D(G).$$ (2) If $|G|$ is odd, then $$\mathcal I_{(n-1,1)}(G)= \mathcal I_{(2,1^{n-2})}(G)=0,$$ and if $|G|\equiv 2\pmod 4$, then $$\mathcal I_{(n-1,1)}(G)=\mathcal P(G)\quad \text{and}\quad \mathcal I_{(2,1^{n-2})}(G)=\mathcal D(G).$$ (3) If $|G|$ is odd and $|G|\ge 7$, then $$ \text{imm}_{(4,1^{n-4})}(\mathcal M_G)=\text{imm}_{(2,2,2,1^{n-6})}(\mathcal M_G).$$

math.CO↗

arXiv2026

DynSplit-KV: Dynamic Semantic Splitting for KVCache Compression in Efficient Long-Context LLM Inference

Although Key-Value (KV) Cache is essential for efficient large language models (LLMs) inference, its growing memory footprint in long-context scenarios poses a significant bottleneck, making KVCache compression crucial. Current compression methods rely on rigid splitting strategies, such as fixed intervals or pre-defined delimiters. We observe that rigid splitting suffers from significant accuracy degradation (ranging from 5.5% to 55.1%) across different scenarios, owing to the scenario-dependent nature of the semantic boundaries. This highlights the necessity of dynamic semantic splitting to match semantics. To achieve this, we face two challenges. (1) Improper delimiter selection misaligns semantics with the KVCache, resulting in 28.6% accuracy loss. (2) Variable-length blocks after splitting introduce over 73.1% additional inference overhead. To address the above challenges, we propose DynSplit-KV, a KVCache compression method that dynamically identifies delimiters for splitting. We propose: (1) a dynamic importance-aware delimiter selection strategy, improving accuracy by 49.9%. (2) A uniform mapping strategy that transforms variable-length semantic blocks into a fixed-length format, reducing inference overhead by 4.9x. Experiments show that DynSplit-KV achieves the highest accuracy, 2.2x speedup compared with FlashAttention and 2.6x peak memory reduction in long-context scenarios.

cs.LG↗

arXiv2023

New constructions of NMDS self-dual codes

Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of $q$-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square $q$, almost $q/8$ NMDS self-dual $q$-ary codes can be constructed.

math.CO↗

arXiv2022

A reciprocity on finite abelian groups involving zero-sum sequences II

Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let $\varphi_G(d)$ be the number of elements in $G$ of order $d$ and $\mathsf M(G,m)$ be the set of all zero-sum sequences of length $m$. In this paper, for any finite abelian group $H$, we prove that $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|$$ if and only if $\varphi_G(d)=\varphi_H(d)$ for any $d|(|G|,|H|)$. We also consider an extension of this result to non-abelian groups in terms of invariant theory.

math.CO↗

arXiv2021

On $*$-clean group rings over finite fields

A ring $R$ is called clean if every element of $R$ is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Va\v{s} introduced a more natural concept of cleanness for $*$-rings, called the $*$-cleanness. More precisely, a $*$-ring $R$ is called a $*$-clean ring if every element of $R$ is the sum of a unit and a projection ($*$-invariant idempotent). Let $\mathbb F$ be a finite field and $G$ a finite abelian group. In this paper, we introduce two classes of involutions on group rings of the form $\mathbb FG$ and characterize the $*$-cleanness of these group rings in each case. When $*$ is taken as the classical involution, we also characterize the $*$-cleanness of $\mathbb F_qG$ in terms of LCD abelian codes and self-orthogonal abelian codes in $\mathbb F_qG$.

math.RA↗

arXiv2021

On cleanness of von Neumann algebras

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.

math.OA↗

arXiv2019

A reciprocity on finite abelian groups involving zero-sum sequences

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the following reciprocity $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|.$$ Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version.

math.CO↗

arXiv2018

On generalized Erd\H{o}s-Ginzburg-Ziv constants of $C_n^r$

Let $G$ be an additive finite abelian group with exponent $\exp(G)=m$. For any positive integer $k$, the $k$-th generalized Erd\H{o}s-Ginzburg-Ziv constant $\mathsf s_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $km$. It is easy to see that $\mathsf s_{kn}(C_n^r)\ge(k+r)n-r$ where $n,r\in\mathbb N$. Kubertin conjectured that the equality holds for any $k\ge r$. In this paper, we mainly prove the following results: (1) For every positive integer $k\ge 6$, we have $$\mathsf s_{kn}(C_n^3)=(k+3)n+O(\frac{n}{\ln n}).$$ (2) For every positive integer $k\ge 18$, we have $$\mathsf s_{kn}(C_n^4)=(k+4)n+O(\frac{n}{\ln n}).$$ (3) For $n\in \mathbb N$, assume that the largest prime power divisor of $n$ is $p^a$ for some $a\in\mathbb N$. For any fixed $r\ge 5$, if $p^t\ge r$ for some $t\in\mathbb N$, then for any $k\in\mathbb N$ we have $$\mathsf s_{kp^tn}(C_n^r)\le(kp^t+r)n+c_r\frac{n}{\ln n},$$ where $c_r$ is a constant depends on $r$. Note that the main terms in our results are consistent with the conjectural values proposed by Kubertin.

math.CO↗

arXiv2017

Erd\H{o}s-Ginzburg-Ziv theorem and Noether number for $C_m\ltimes_{\varphi} C_{mn}$

Let $G$ be a multiplicative finite group and $S=a_1\cdot\ldots\cdot a_k$ a sequence over $G$. We call $S$ a product-one sequence if $1=\prod_{i=1}^ka_{\tau(i)}$ holds for some permutation $\tau$ of $\{1,\ldots,k\}$. The small Davenport constant $\mathsf d(G)$ is the maximal length of a product-one free sequence over $G$. For a subset $L\subset \mathbb N$, let $\mathsf s_L(G)$ denote the smallest $l\in\mathbb N_0\cup\{\infty\}$ such that every sequence $S$ over $G$ of length $|S|\ge l$ has a product-one subsequence $T$ of length $|T|\in L$. Denote $\mathsf e(G)=\max\{\text{ord}(g): g\in G\}$. Some classical product-one (zero-sum) invariants including $\mathsf D(G):=\mathsf s_{\mathbb N}(G)$ (when $G$ is abelian), $\mathsf E(G):=\mathsf s_{\{|G|\}}(G)$, $\mathsf s(G):=\mathsf s_{\{\mathsf e(G)\}}(G)$, $\eta(G):=\mathsf s_{[1,\mathsf e(G)]}(G)$ and $\mathsf s_{d\mathbb N}(G)$ ($d\in\mathbb N$) have received a lot of studies. The Noether number $\beta(G)$ which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let $G\cong C_m\ltimes_{\varphi} C_{mn}$, in this paper, we prove that $$\mathsf E(G)=\mathsf d(G)+|G|=m^2n+m+mn-2$$ and $\beta(G)=\mathsf d(G)+1=m+mn-1$. We also prove that $\mathsf s_{mn\mathbb N}(G)=m+2mn-2$ and provide the upper bounds of $\eta(G)$, $\mathsf s(G)$. Moreover, if $G$ is a non-cyclic nilpotent group and $p$ is the smallest prime divisor of $|G|$, we prove that $\beta(G)\le \frac{|G|}{p}+p-1$ except if $p=2$ and $G$ is a dicyclic group, in which case $\beta(G)=\frac{1}{2}|G|+2$.

math.CO↗