The fourth generalized Davenport constant of C_5^3

Abstract

For a finite abelian group G and k ≥ 1, the generalized Davenport constant D_k(G) is the least ℓ such that every sequence over G of length at least ℓ has k pairwise disjoint nonempty zero-sum subsequences. A theorem of Freeze and Schmid gives D_k(C₅³) ≥ 5k+10 for every k ≥ 2. We prove the matching upper bound: D₄(C₅³)=30, and hence D_k(C₅³)=5k+10 for every k ≥ 2, so the Freeze--Schmid bound is attained by C₅³ from k=2 onward, as it is by C₂³ and unlike C₃³. The proof is finite and computer-assisted. The remaining case reduces to showing that every zero-sum sequence of length $31$ over C₅³ contains a nonempty zero-sum subsequence of length at most five. A saturation argument confines the multiplicities of a hypothetical counterexample to {1,2,4}, its support pattern to one of $60$ solutions of two linear equations, and its geometry to one of $78$ rank/plane branches normalized to a standard basis; an exhaustive search exhausts every branch with no survivor. The search was carried out by three independently written implementations, and the branch cover was regenerated by separate programs from the lemmas alone; two further machine-verified values, D₃(C₅³)=25 and s_{≤ 6}(C₅³)=24, enter the second statement, and their records accompany the paper.

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BibTeX RIS

BibTeX

@misc{yiu2026fourth,
  title = {{The fourth generalized Davenport constant of $C_5^3$}},
  author = {Sze Chun Yiu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04950},
  eprint = {2609.04950},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04950},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The fourth generalized Davenport constant of C_5^3
AU  - Sze Chun Yiu
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04950
UR  - https://arxiv.org/abs/2609.04950
AB  - For a finite abelian group G and k ≥ 1, the generalized Davenport constant D_k(G) is the least ℓ such that every sequence over G of length at least ℓ has k pairwise disjoint nonempty zero-sum subsequences. A theorem of Freeze and Schmid gives D_k(C_5^3) ≥ 5k+10 for every k ≥ 2. We prove the matching upper bound: D_4(C_5^3)=30, and hence D_k(C_5^3)=5k+10 for every k ≥ 2, so the Freeze--Schmid bound is attained by C_5^3 from k=2 onward, as it is by C_2^3 and unlike C_3^3. The proof is finite and computer-assisted. The remaining case reduces to showing that every zero-sum sequence of length $31$ over C_5^3 contains a nonempty zero-sum subsequence of length at most five. A saturation argument confines the multiplicities of a hypothetical counterexample to {1,2,4}, its support pattern to one of $60$ solutions of two linear equations, and its geometry to one of $78$ rank/plane branches normalized to a standard basis; an exhaustive search exhausts every branch with no survivor. The search was carried out by three independently written implementations, and the branch cover was regenerated by separate programs from the lemmas alone; two further machine-verified values, D_3(C_5^3)=25 and s_≤ 6(C_5^3)=24, enter the second statement, and their records accompany the paper.
ER  -

The fourth generalized Davenport constant of C₅³

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