arXiv subjects

Qinghai Zhong

Publications and source records attributed to Qinghai Zhong.

At least 37 records · Page 2Linked to original sources

arXiv2019

On Erdős-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups

Let $G$ be a finite group and exp$(G)$ = lcm$\{$ord$(g)$$\mid$$g \in G \}$. A finite unordered sequence of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. We denote by $\mathsf s (G)$ (or $\mathsf E (G)$ respectively) the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $\exp (G)$ (or $|G|$ respectively). In this paper, we provide the exact values of $\mathsf s (G)$ and $\mathsf E (G)$ for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length $\mathsf s (G) - 1$ (or $\mathsf E (G) - 1$ respectively) having no product-one subsequence of length $\exp (G)$ (or $|G|$ respectively).

math.CO↗

arXiv2019

On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.

math.CO↗

arXiv2018

On the arithmetic of Mori monoids and domains

Let $R$ be a Mori domain with complete integral closure $\widehat R$, nonzero conductor $\mathfrak f = (R \ :\ \widehat R)$, and suppose that both $v$-class groups $\mathcal C_v (R)$ and $\mathcal C_v (\widehat R)$ are finite. If $R/\mathfrak f$ is finite, then the elasticity of $R$ is either rational or infinite. If $R/\mathfrak f$ is artinian, then unions of sets of lengths of $R$ are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of $v$-noetherian monoids.

math.AC↗

arXiv2018

A characterization of seminormal C-monoids

It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class group of Krull monoids. We prove that a C-monoid is seminormal if and only if its reduced class semigroup is a union of groups. Based on this characterization we establish a criterion (in terms of the class semigroup) when seminormal C-monoids are half-factorial.

math.AC↗

arXiv2018

On elasticities of locally finitely generated monoids

Let $H$ be a commutative and cancellative monoid. The elasticity $ρ(a)$ of a non-unit $a \in H$ is the supremum of $m/n$ over all $m, n$ for which there are factorizations of the form $a=u_1 \cdot \ldots \cdot u_m=v_1 \cdot \ldots \cdot v_{n}$, where all $u_i$ and $v_j$ are irreducibles. The elasticity $ρ(H)$ of $H$ is the supremum over all $ρ(a)$. We establish a characterization, valid for finitely generated monoids, when every rational number $q$ with $1< q < ρ(H)$ can be realized as the elasticity of some element $a \in H$. Furthermore, we derive results of a similar flavor for locally finitely generated monoids (they include all Krull domains and orders in Dedekind domains satisfying certain algebraic finiteness conditions) and for weakly Krull domains.

math.AC↗

arXiv2018

Sets of Arithmetical Invariants in Transfer Krull Monoids

Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between $1$ and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.

math.AC↗

arXiv2018

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Let $H$ be a transfer Krull monoid over a finite ablian group $G$ (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit $a \in H$ can be written as a product of irreducible elements, say $a = u_1 \ldots u_k$, and the number of factors $k$ is called the length of the factorization. The set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the group $G$. Let $H'$ be a further transfer Krull monoid over a finite ablian group $G'$ and suppose that $\mathcal L (H)= \mathcal L (H')$. We prove that, if $G\cong C_n^r$ with $r\le n-3$ or ($r\ge n-1\ge 2$ and $n$ is a prime power), then $G$ and $G'$ are isomorphic.

math.NT↗

arXiv2017

Long sets of lengths with maximal elasticity

We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $Δ_ρ (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $ρ(H)$. We study $Δ_ρ (H)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

math.AC↗

arXiv2017

Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids

Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.

math.AC↗

arXiv2016

Sets of minimal distances and characterizations of class groups of Krull monoids

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. There is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in Δ^* (H)$, where $Δ^* (H)$ denotes the set of minimal distances of $H$. We study the structure of $Δ^* (H)$ and establish a characterization when $Δ^*(H)$ is an interval. The system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the class group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to $C_n^r$ with $r,n \in \mathbb N$ and $Δ^*(H)$ is not an interval.

math.AC↗

arXiv2016

The set of distances in seminormal weakly Krull monoids

The set of distances of a monoid or of a domain is the set of all $d \in \mathbb N$ with the following property: there are irreducible elements $u_1, \ldots, u_k, v_1, \ldots, v_{k+d}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{k+d}$, but $u_1 \cdot \ldots \cdot u_k$ cannot be written as a product of $l$ irreducible elements for any $l$ with $k < l < k+d$. We show that the set of distances is an interval for certain seminormal weakly Krull monoids which include seminormal orders in holomorphy rings of global fields.

math.AC↗

arXiv2015

Products of $k$ atoms in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\in \mathbb N$, let $\mathcal U_k(H)$ denote the set of all $m\in \mathbb N$ with the following property: There exist atoms $u_1, \ldots, u_k, v_1, \ldots , v_m\in H$ such that $u_1\cdot\ldots\cdot u_k=v_1\cdot\ldots\cdot v_m$. It is well-known that the sets $\mathcal U_k (H)$ are finite intervals whose maxima $ρ_k(H)=\max \mathcal U_k(H) $ depend only on $G$. If $|G|\le 2$, then $ρ_k (H) = k$ for every $k \in \mathbb N$. Suppose that $|G| \ge 3$. An elementary counting argument shows that $ρ_{2k}(H)=k\mathsf D(G)$ and $k\mathsf D(G)+1\le ρ_{2k+1}(H)\le k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ where $\mathsf D(G)$ is the Davenport constant. In \cite{Ga-Ge09b} it was proved that for cyclic groups we have $k\mathsf D(G)+1 = ρ_{2k+1}(H)$ for every $k \in \mathbb N$. In the present paper we show that (under a mild condition on the Davenport constant) for every noncyclic group there exists a $k^*\in \mathbb N$ such that $ρ_{2k+1}(H)= k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ for every $k\ge k^*$. This confirms a conjecture of A. Geroldinger, D. Grynkiewicz, and P. Yuan in \cite{Ge-Gr-Yu15}.

math.AC↗

arXiv2015

The set of minimal distances in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. If $G$ is finite, then there is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in Δ^* (H)$, where $Δ^* (H)$ denotes the set of minimal distances of $H$. We show that $\max Δ^* (H) \le \max \{\exp (G)-2, \mathsf r (G)-1\}$ and that equality holds if every class of $G$ contains a prime divisor, which holds true for holomorphy rings in global fields.

math.AC↗

arXiv2015

A Characterization of class groups via sets of lengths {II}

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. If an element $a \in H$ has a factorization $a=u_1 \cdot \ldots \cdot u_k$ into irreducible elements $u_1, \ldots, u_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the class group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the class group. We verify the conjecture if the class group is isomorphic to $C_n^r$ with $r,n \ge 2$ and $r \le \max \{2, (n+2)/6\}$. Indeed, let $H'$ be a further Krull monoid with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H)= \mathcal L (H')$. We prove that, if one of the groups $G$ and $G'$ is isomorphic to $C_n^r$ with $r,n$ as above, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).

math.NT↗

arXiv2015

On the Erd{ő}s-Ginzburg-Ziv constant of groups of the form $C_2^r\oplus C_n$

Let $G$ be a finite abelian group. The Erd{ő}s-Ginzburg-Ziv constant $\mathsf s(G)$ of $G$ is defined as the smallest integer $l\in \mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a zero-sum subsequence $T$ of length $|T|= {\exp}(G)$. The value of this classical invariant for groups with rank at most two is known. But the precise value of $\mathsf s(G)$ for the groups of rank larger than two is difficult to determine. In this paper we pay our attentions to the groups of the form $C_2^{r-1}\oplus C_{2n}$, where $r\geq 3$ and $n\ge 2$. We give a new upper bound of $\mathsf s(C_2^{r-1}\oplus C_{2n})$ for odd integer $n$. For $r\in [3,4]$, we obtain that $\mathsf s(C_2^2\oplus C_{2n})=4n+3$ for $n\ge 2$ and $\mathsf s(C_2^{3}\oplus C_{2n})=4n+5$ for $n\geq 36$.

math.CO↗

arXiv2014

The catenary degree of Krull monoids II

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G| \ge 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf c (H) \in [3, \mathsf D (G)]$, where $\mathsf D (G)$ denotes the Davenport constant of $G$. It is well-known when $\mathsf c (H) \in \{3,4, \mathsf D (G)\}$ holds true. Based on a characterization of the catenary degree determined in the first paper (The catenary degree of Krull monoids I), we determine the class groups satisfying $\mathsf c (H)= \mathsf D (G)-1$. Apart from the mentioned extremal cases the precise value of $\mathsf c (H)$ is known for no further class groups.

math.AC↗

arXiv2012

On the maximal multiplicity of long zero-sum free sequences over $C_p\oplus C_p$

In this paper, we point out that the method used in [Acta Arith. 128(2007) 245-279] can be modified slightly to obtain the following result. Let $\varepsilon \in (0,\frac 14)$ and $c>0$, and let $p$ be a sufficiently large prime depending on $\varepsilon$ and $c$. Then every zero-sumfree sequence $S$ over $C_p\oplus C_p$ of length $|S|\geq 2p-c\sqrt{p}$ contains some element at least $\lfloor p^{\frac14-\varepsilon}\rfloor$ times.

math.NT↗

arXiv2011

On short zero-sum subsequences of zero-sum sequences

Let $G$ be a finite abelian group, and let $η(G)$ be the smallest integer $d$ such that every sequence over $G$ of length at least $d$ contains a zero-sum subsequence $T$ with length $|T|\in [1,\exp(G)]$. In this paper, we investigate the question whether all non-cyclic finite abelian groups $G$ share with the following property: There exists at least one integer $t\in [\exp(G)+1,η(G)-1]$ such that every zero-sum sequence of length exactly $t$ contains a zero-sum subsequence of length in $[1,\exp(G)]$. Previous results showed that the groups $C_n^2$ ($n\geq 3$) and $C_3^3$ have the property above. In this paper we show that more groups including the groups $C_m\oplus C_n$ with $3\leq m\mid n$, $C_{3^a5^b}^3$, $C_{3\times 2^a}^3$, $C_{3^a}^4$ and $C_{2^b}^r$ ($b\geq 2$) have this property. We also determine all $t\in [\exp(G)+1, η(G)-1]$ with the property above for some groups including the groups of rank two, and some special groups with large exponent.

math.NT↗