  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.
