  We generalize the notion of Erd\H{o}s-Ginzburg-Ziv constants -- along the
same lines we generalized in earlier work the notion of Davenport constants --
to a ``higher degree" and obtain various lower and upper bounds. These bounds
are sometimes exact as is the case for certain finite commutative rings of
prime power cardinality. We also consider to what extent a theorem due
independently to W.D.~Gao and the first author that relates these two
parameters extends to this higher degree setting. Two simple examples that
capture the essence of these higher degree Erd\H{o}s-Ginzburg-Ziv constants are
the following. 1) Let $\nu_p(m)$ denote the $p-$adic valuation of the integer
$m$. Suppose we have integers $t | {m \choose 2}$ and $n=t+2^{\nu_2(m)}$, then
every sequence $S$ over ${\mathbb Z}_2$ of length $|S| \geq n$ contains a
subsequence $S'$ of length $t$ for which $\sum_{a_{i_1},\ldots, a_{i_m} \in S'}
a_{i_1}\cdots a_{i_m} \equiv 0 \pmod{2}$, and this is sharp. 2) Suppose
$k=3^{\alpha}$ for some integer $\alpha \geq 2$. Then every sequence $S$ over
${\mathbb Z}_3$ of length $|S| \geq k+6$ contains a subsequence $S'$ of length
$k$ for which $\sum_{a_h, a_i, a_j \in S'} a_ha_ia_j \equiv 0 \pmod{3}$. These
examples illustrate that if a sequence of elements from a finite commutative
ring is long enough, certain symmetric expressions (symmetric polynomials) have
to vanish on the elements of a subsequence of prescribed length. The
Erd\H{o}s-Ginzburg-Ziv Theorem is just the case where a sequence of length
$2n-1$ over ${\mathbb Z}_n$ contains a subsequence $S'=(a_1, \ldots, a_n)$ of
length $n$ that vanishes when substituted in the linear symmetric polynomial
$a_1+\cdots+a_n.$
