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Article Dans Une Revue Integers : Electronic Journal of Combinatorial Number Theory Année : 2021

Higher degree Davenport constants over finite commutative rings

Résumé

We generalize the notion of Davenport constants to a 'higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples that capture the essence of these higher degree Davenport constants are the following. 1) Suppose $n = 2^k$, then every sequence of integers $S$ of length $2n$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$ and the bound is sharp. 2) Suppose $n \equiv1 \pmod{2}$, then every sequence of integers $S$ of length $2n -1$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$. These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions have to vanish on the elements of a subsequence.
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Dates et versions

hal-03139286 , version 1 (11-02-2021)
hal-03139286 , version 2 (14-02-2022)

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  • HAL Id : hal-03139286 , version 2

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Yair Caro, Benjamin Girard, John R. Schmitt. Higher degree Davenport constants over finite commutative rings. Integers : Electronic Journal of Combinatorial Number Theory, 2021, 21, pp.A120. ⟨hal-03139286v2⟩
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