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arXiv 2207.08682math.COmath.ACmath.NT
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高次Erdos-Ginzburg-Ziv常数

Higher Degree Erdos-Ginzburg-Ziv Constants

Yair Caro, John R. Schmitt

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摘要：

我们将Erdens-Ginzburg-Ziv常数的概念推广到“更高的程度”，并得到各种上下界。这些界限有时是精确的，就像某些素数幂基数的有限交换环一样。我们还考虑了独立于W. D.~ Gao和第一作者将这两个参数联系起来，扩展到这个更高程度的设置。下面是两个简单的例子，它们抓住了这些高阶Erdens-Ginzburg-Ziv常数的本质。1)令$ν_p(m)$表示整数$m$的$p-$进赋值。假设我们有整数$t | {m \choose 2}$和$n=t+2^{ν_2(m)}$，那么每个长度为$|S| \geq n$的序列$S$在${\mathbb Z}_2$上包含一个长度为$t$的子序列$S'$，其中$\sum_{a_{i_1},\ldots, a_{i_m} \in S'} a_{i_1}\cdots a_{i_m} \equiv 0 \pmod{2}$，这是尖锐的。2)假设$k=3^α$为某个整数$α\geq 2$。则长度为$|S| \geq k+6$的${\mathbb Z}_3$上的每个序列$S$包含长度为$k$的子序列$S'$，其中$\sum_{a_h, a_i, a_j \in S'} a_ha_ia_j \equiv 0 \pmod{3}$。这些例子说明，如果一个有限交换环的元素序列足够长，某些对称表达式（对称多项式）必须在指定长度的子序列的元素上消失。Erdens-Ginzburg-Ziv定理就是这样一种情况，其中${\mathbb Z}_n$上长度为$2n-1$的序列包含长度为$n$的子序列$S'=(a_1, \ldots, a_n)$，该子序列在被线性对称多项式$a_1+\cdots+a_n.$替换时消失

英文摘要：

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 $a_1+\cdots+a_n.$

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