  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 $\sum_{a_i,a_j \in
S'} a_ia_j =0 \pmod{n}$ and the bound is sharp.
  2) Suppose $n =1 \pmod{2}$, then every sequence of integers $S$ of length $2n
-1$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j
\in S'} a_ia_j =0 \pmod{n}$.
  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.
