We generalize the notion of Erdős-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 Erdős-Ginzburg-Ziv constants are the following. 1) Let $ν_p(m)$ denote the $p-$adic valuation of the integer $m$. Suppose we have integers $t | {m \choose 2}$ and $n=t+2^{ν_2(m)}$, 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 $k=3^α$ for some integer $α\geq 2$. 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 Erdős-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 $a_1+\cdots+a_n.$