On the Davenport constant and group algebras

• Autorzy: Daniel Smertnig
• Języki publikacji: EN

Abstrakty

EN

For a finite abelian group G and a splitting field K of G, let 𝖽(G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g₁ · ... · g_{l}$ over G such that $(X^{g₁} - a₁) · ... · (X^{g_{l}} - a_{l}) ≠ 0 ∈ K[G]$ for all $a₁, ..., a_{l} ∈ K^{×}$. If 𝖣(G) denotes the Davenport constant of G, then there is the straightforward inequality 𝖣(G) - 1 ≤ 𝖽(G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which 𝖣(G) - 1 < 𝖽(G,K). Thus we disprove the conjecture.

Kategorie tematyczne

• 11P70: Inverse problems of additive number theory, including sumsets
• 11B30: Arithmetic combinatorics; higher degree uniformity
• 11B50: Sequences (mod m )
• 20K01: Finite abelian groups [For sumsets, see and ]

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2010
• Tom: 121
• Numer: 2
• Strony: 179-193

Daty

wydano

2010

Twórcy

autor

Daniel Smertnig

• Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria

Identyfikatory

DOI

10.4064/cm121-2-2