On the Davenport constant and group algebras

Daniel Smertnig. "On the Davenport constant and group algebras." Colloquium Mathematicae 121.2 (2010): 179-193. <http://eudml.org/doc/286476>.

@article{DanielSmertnig2010,
abstract = {For a finite abelian group G and a splitting field K of G, let (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 (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (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 (G) - 1 < (G,K). Thus we disprove the conjecture.},
author = {Daniel Smertnig},
journal = {Colloquium Mathematicae},
language = {eng},
number = {2},
pages = {179-193},
title = {On the Davenport constant and group algebras},
url = {http://eudml.org/doc/286476},
volume = {121},
year = {2010},
}

TY - JOUR
AU - Daniel Smertnig
TI - On the Davenport constant and group algebras
JO - Colloquium Mathematicae
PY - 2010
VL - 121
IS - 2
SP - 179
EP - 193
AB - For a finite abelian group G and a splitting field K of G, let (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 (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (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 (G) - 1 < (G,K). Thus we disprove the conjecture.
LA - eng
UR - http://eudml.org/doc/286476
ER -