Higher Degree Davenport Constants over Finite Commutative Rings

TL;DR

This paper extends the concept of Davenport constants to higher degrees over finite commutative rings, providing bounds and exact values for specific cases, and demonstrating that long sequences contain subsequences with vanishing symmetric sums.

Contribution

It introduces the higher degree Davenport constants over finite rings, establishes bounds, and provides exact results for prime power cardinality rings.

Findings

Exact bounds for certain finite rings of prime power size.

Examples demonstrating the existence of subsequences with zero symmetric sums.

Bounds are sharp in specific cases.

Abstract

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.