Davenport Constant Research Articles

Product of polynomial values being large power

Abstract Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

The universal zero-sum invariant and weighted zero-sum for infinite abelian groups

Let G be an abelian group, and let F ( G ) be the free commutative monoid with basis G. For Ω ⊂ F ( G ) , define the universal zero-sum invariant d Ω ( G ) to be the smallest integer l such that every sequence T over G of length l has a subsequence in Ω . The invariant d Ω ( G ) unifies many classical zero-sum invariants. Let B ( G ) be the submonoid of F ( G ) consisting of all zero-sum sequences over G, and let A ( G ) be the set consisting of all minimal zero-sum sequences over G. The empty sequence, which is the identity of B ( G ) , is denoted by ε . The well-known Davenport constant D ( G ) of the group G can be also represented as d B ( G ) ∖ { ε } ( G ) or d A ( G ) ( G ) in terms of the universal zero-sum invariant. Notice that A ( G ) is the unique minimal generating set of the monoid B ( G ) from the point of view of Algebra. Hence, it would be interesting to determine whether A ( G ) is minimal to represent the Davenport constant or not for a general finite abelian group G. In this paper, we show that except for a few special classes of groups, there always exists a proper subset Ω of A ( G ) such that d Ω ( G ) = D ( G ) . Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of infinite abelian groups. The universal zero-sum invariant d Ω ; Ψ ( G ) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant D Ψ ( G ) (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that D Ψ ( G ) < ∞ in terms of the weights set Ψ when | Ψ | is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group G by cosets of some given subgroups of G, and the finiteness of weighted Davenport constant.

Quadratic symmetric polynomials and an analogue of the Davenport constant

We define the constant $$ D(\varphi_F, p) $$ , an analogue for the Davenport constant, for sequences on the finite field $$ \mathbb{F}_p $$ , defined via quadratic symmetric polynomials. Next, we state a series of results presenting either the exact value of $$ D(\varphi_F, p) $$ , or lower and upper bounds for this constant.

A note on Bass' conjecture

For a finite group G, we denote by d(G) and by E(G), respectively, the small Davenport constant and the Gao constant of G. Let Cn be the cyclic group of order n and let Gm,n,s=Cn⋊sCm be a metacyclic group. In [2, Conjecture 17], Bass conjectured that d(Gm,n,s)=m+n−2 and E(Gm,n,s)=mn+m+n−2 provided ordn(s)=m. In this paper, we show that the assumption ordn(s)=m is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for Gm,n,s and the mn-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2≡s(modn).

On Davenport Constant of the Group $C_2^{r-1} \oplus C_{2k}$

Let $G$ be a finite abelian group. The Davenport constant $\mathsf{D}(G)$ is the maximal length of minimal zero-sum sequences over $G$. For groups of the form $C_2^{r-1} \oplus C_{2k}$ the Davenport constant is known for $r\leq 5$. In this paper, we get the precise value of $\mathsf{D}(C_2^{5} \oplus C_{2k})$ for $k\geq 149$. It is also worth pointing out that our result can imply the precise value of $\mathsf{D}(C_2^{4} \oplus C_{2k})$.

The Weighted Davenport constant of a group and a related extremal problem-II

For a finite abelian group G with exp(G)=n and an integer k≥2, Balachandran and Mazumdar (2019) introduced the extremal function fG(D)(k) which is defined to be min{|A|:0̸≠A⊆[1,n−1] withDA(G)≤k} (and ∞ if there is no such A), where DA(G) denotes the A-weighted Davenport constant of the group G. Denoting fG(D)(k) by f(D)(p,k) when G=Fp (for p prime), it is known (Balachandran and Mazumdar, 2019) that p1/k−1≤f(D)(p,k)≤O(plogp)1/k holds for each k≥2 and p sufficiently large, and that for k=2,4, we have the sharper bound f(D)(p,k)≤O(p1/k). It was furthermore conjectured that f(D)(p,k)=Θ(p1/k). In this short paper we prove that f(D)(p,k)≤4k2p1/k for sufficiently large primes p.

On the incomparability of systems of sets of lengths

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. We consider the system L(H) of all sets of lengths of H and study when L(H) contains or is contained in a system L(H′) of a Krull monoid H′ with finite class group G′, prime divisors in all classes and Davenport constant D(G′)=D(G). Among others, we show that if G is either cyclic of order m≥7 or an elementary 2-group of rank m−1≥6, and G′ is any group which is non-isomorphic to G but with Davenport constant D(G′)=D(G), then the systems L(H) and L(H′) are incomparable.

Extremal product-one free sequences over Cn⋊sC2

Let G be a finite group multiplicatively written. The small Davenport constant of G is the maximum positive integer d(G) such that there exists a product-one free sequence S of length d(G). Let s2≡1(modn), where s≢±1(modn). It has been proven that d(Cn⋊sC2)=n (see [13, Lemma 6]). In this paper, we determine all sequences over Cn⋊sC2 of length n which are product-one free. It completes the classification of all product-one free sequences over every group of the form Cn⋊sC2, including the quasidihedral groups and the modular maximal-cyclic groups.

Extremal product-one free sequences and |G|-product-one free sequences of a metacyclic group

Let G be a multiplicatively written finite group. We denote by E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that E(G)≤2|G|−1 for every finite abelian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that E(G)=d(G)+|G| for every finite group, where d(G) is the small Davenport constant. Very recently, we confirmed this conjecture for the case when G=〈x,y|xp=ym=1,x−1yx=yr〉 where p is the smallest prime divisor of |G| and gcd(p(r−1),m)=1. In this paper, we study the associated inverse problems on d(G) and E(G). Our main results characterize the structure of any product-one free sequence with extremal length d(G), and that of any |G|-product-one free sequence with extremal length E(G)−1.

On a conjecture of the small Davenport constant for finite groups

Let G be a multiplicatively written finite group. We denote by d(G) the small Davenport constant of G, that is, the maximal integer ℓ such that there is a sequence of length ℓ over G which has no non-trivial product-one subsequence. In 2014, Gao, Li, and Peng conjectured that d(G)≤|G|/p+p−2 for any finite non-cyclic group G, where p is the smallest prime divisor of |G|. In this paper, we confirm that this conjecture is true.

Erdős-Burgess constant of commutative semigroups

Let be a nonempty commutative semigroup written additively. An element e of is said to be idempotent if e + e = e. The Erdős-Burgess constant of the semigroup is defined as the smallest positive integer such that any -valued sequence T of length contains a nonempty subsequence the sum of whose terms is an idempotent of We make a study of when is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that is finite, and we obtain sharp bounds of in case is finite, and determine the precise value of in some cases which unifies some well known results on the precise value of Davenport constant in the setting of commutative semigroups.

Davenport constant of a box in $\mathbb {Z}^2$

Davenport constant of a box in $\mathbb {Z}^2$

On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right

Let $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$, denoted by $\textsf{D}(X)$, is defined to be the supremum of lengths of all minimal zero-sum sequences over $X$. In this paper, we study the minimal zero-sum sequences over $X = \left[\kern-0.15em\left[ { - 1, 1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m, n} \right]\kern-0.15em\right]\subset\mathbb{Z}^2$. We completely determine the structure of minimal zero-sum sequences of maximal length over $X$ and obtain that $\textsf{D}(X) = 2(n+m).$

Correction to: Davenport constant for semigroups

Correction to: Davenport constant for semigroups

A generalization of Kruyswijk-Olson theorem on Davenport constant in commutative semigroups

Let $\mathcal{S}$ be a finite commutative semigroup written additively. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e = e$. The Erdős-Burgess constant of the semigroup $\mathcal{S}$ is defined as the smallest positive integer $\ell$ such that any $\mathcal{S}$-valued sequence $T$ of length $\ell$ must contain one or more terms with the sum being an idempotent of $\mathcal{S}$. If the semigroup $\mathcal{S}$ is a finite abelian group, the Erdős-Burgess constant reduces to the well-known Davenport constant in Combinatorial Number Theory. In this paper, we determine the value of the Erdős-Burgess constant for a direct sum of two finite cyclic semigroups in some cases, which generalizes the classical Kruyswijk-Olson Theorem on Davenport constant of finite abelian groups in the setting of commutative semigroups.

A stronger connection between the Erdős-Burgess and Davenport constants

The Erdős-Burgess constant of a semigroup S is the smallest positive integer k such that any sequence over S of length k contains a nonempty subsequence whose terms multiply to an idempotent element of S. In the case where S is the multiplicative semigroup of Z/nZ, we confirm a conjecture connecting the Erdős-Burgess constant of S and the Davenport constant of (Z/nZ)× for n with at most two prime factors. We also discuss the extension of our techniques to other rings.

On the lower bounds of Davenport constant

Let G=Cn1⊕…⊕Cnr with 1<n1|…|nr be a finite abelian group. The Davenport constant D(G) is the smallest integer t such that every sequence S over G of length |S|≥t has a non-empty zero-sum subsequence. It is a starting point of zero-sum theory. It has a trivial lower bound D⁎(G)=n1+…+nr−r+1, which equals D(G) over p-groups. We investigate the non-dispersive sequences over groups Cnr, thereby revealing the growth of D(G)−D⁎(G) over non-p-groups G=Cnr⊕Ckn with n,k≠1. We give a general lower bound of D(G) over non-p-groups and show that if G is an abelian group with exp⁡(G)=m and rank r, fix m>0 a non-prime-power, then for each N>0 there exists an ε>0 such that if |G|/mr<ε, then D(G)−D⁎(G)>N.

Zero-sum invariants on finite abelian groups with large exponent

Let G be an additive finite abelian group with exponent n. Let D(G) be the Davenport constant of G, skn(G) the kth Erdős–Ginzburg–Ziv constant of G, where k is a positive integer. Recently, Gao, Han, Peng and Sun conjectured that skn(G)=kn+D(G)−1 holds if k≥⌈D(G)n⌉. Let m,n be positive integers and H an abelian p-group with D(H)≤pn. Let G=H⊕Cmpn. For any integer k≥2, we prove that skmpn(G)=(k+1)mpn+D(H)−2=kmpn+D(G)−1. This verifies the above conjecture in this case. We also provide asymptotically tight bounds for zero-sum invariants D(G), skn(G) and η(G) for a class of abelian groups with large exponent.

Extremal product-one free sequences in Cq⋊sCm

Let G be a finite group. The Davenport constant of G is the smallest positive integer d such that every sequence over G with d elements has a non-empty subsequence with product 1. Let Cn be the cyclic group of order n. Bass [1] showed that the Davenport constant of the metacyclic group Cq⋊sCm, where q is a prime number and ordq(s)=m≥2, is m+q−1. In this paper, we determine the form of all sequences S of Cq⋊sCm, with q+m−2 elements that are product-one free.

Minimal zero-sum sequences over 〚 − m,n〛

TextLet X be a subset of an abelian group G. A sequence S over X is called zero-sum if the sum of S is zero, and minimal zero-sum if it is a non-empty zero-sum sequence such that all proper subsequences are not zero-sum. The Davenport constant of X, denoted by D(X), is defined as the supremum of lengths of minimal zero-sum sequences over X. In this paper, we focus on minimal zero-sum sequences over 〚−m,n〛⊂Z. We obtain the structure of minimal zero-sum sequences over 〚−m,n〛 of length at least m+n−min⁡{m,n}+5+3. As a corollary, the Davenport constant D(〚−m,n〛) is determined provided that m and n are both sufficiently large. This together with the previous result of the authors gives the value of D(〚−m,n〛) for all but a finite number of pairs (m,n)∈N2. VideoFor a video summary of this paper, please visit https://youtu.be/iXKynVwXGZc.