Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 23 min ago
PreprintCC BY 4.0arXiv

Extremal zero-sum sequences without elements of maximal order

  • Yongke Qu
  • Guoqing Wang
Open at sourcePDFDOICite

Abstract

Let D(G) denote the Davenport constant of a finite abelian group G, the maximal length of a minimal zero-sum sequence over G. Gao and Geroldinger conjectured that every minimal zero-sum sequence of length D(G)~\cite[Conjecture~6.1]{GG99} and every zero-sumfree sequence of length D(G)−1~\cite[Conjecture~4.1]{GG06} contains an element of order exp(G). We disprove both conjectures by proving a density-one theorem: for every fixed finite abelian group H of exponent M=2apb, where a≥2, b≥1, and p is an odd prime, the set of integers r≥0 for which Gr=C2r⊕ H admits a minimal zero-sum sequence of length D(Gr) with no term of order M has natural density one.

Related

  1. Degree sequences of spectrally extremal connected nonregular graphs with prescribed maximum degree: the dense case
    Zejun Huang, Chenxi Yang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module
    You Zhou
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  3. Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples
    Beata Deregowska, Barbara Lewandowska
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  4. Systems of equations with elements of finitely generated groups and with linear recurrence sequences
    Subham Bhakta, Alina Ostafe, Igor E. Shparlinski
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Abelian maximal pattern complexity and extremal words
    Qingcheng Zeng, Yumei Xue, Cheng Zeng
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsDynamical systemsarXiv

Cite

BibTeXRIS
BibTeX
@misc{qu2026extremal,
  title = {{Extremal zero-sum sequences without elements of maximal order}},
  author = {Yongke Qu and Guoqing Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02063},
  eprint = {2611.02063},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02063},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Extremal zero-sum sequences without elements of maximal order
AU  - Yongke Qu
AU  - Guoqing Wang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02063
UR  - https://arxiv.org/abs/2611.02063
AB  - Let D(G) denote the Davenport constant of a finite abelian group G, the maximal length of a minimal zero-sum sequence over G. Gao and Geroldinger conjectured that every minimal zero-sum sequence of length D(G)~\cite[Conjecture~6.1]{GG99} and every zero-sumfree sequence of length D(G)−1~\cite[Conjecture~4.1]{GG06} contains an element of order exp(G). We disprove both conjectures by proving a density-one theorem: for every fixed finite abelian group H of exponent M=2^ap^b, where a≥2, b≥1, and p is an odd prime, the set of integers r≥0 for which G_r=C_2^r⊕ H admits a minimal zero-sum sequence of length D(G_r) with no term of order M has natural density one.
ER  -