Erdos-Ginzburg-Ziv theorem and Noether number for C-m proportional to(phi) C-mn
Erdos-Ginzburg-Ziv theorem and Noether number for C-m proportional to(phi) C-mn
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Erdos-Ginzburg-Ziv 定理和 C-m 与 (phi) C-mn 成正比的诺特数
DOI：
10.1016/j.jnt.2018.10.007
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发表时间：
2019
影响因子：
0.7
通讯作者：
Zhang Hanbin
中科院分区：
文献类型：
--
作者：
Han Dongchun;Zhang Hanbin
设G是一个乘法有限群，S= a1 <$...<$ak是G上的一个序列.我们称S为积1序列，如果1= 1 i= 1 k a τ（i）对{1，.，k}的某个置换τ成立。小Davenport常数d（G）是G上无积1序列的最大长度。对于子集L <$N，设s L（G）表示最小l∈ N 0 <${∞}，使得G上长度为|S| ≥ l有一个长度为T的积1子序列|不|∈ L.记e（G）= max <${ord（g）：g∈ G}.一些经典的积一（零和）不变量包括D（G）：= sN（G）（当G是阿贝尔的），E（G）：= s {|G|}（G），s（G）：= s {e（G）}（G），η（G）：= s [1，e（G）]（G）和sdN（G）（d∈ N）的研究已经取得了很多成果。与零和理论密切相关的Noether数β（G）被定义为多项式不变量代数生成元的最大次数界。设G <$Cm <$φ Cm n，本文证明了E（G）= d（G）+|G| = m 2 n+ m+ m n− 2和β（G）= d（G）+ 1= m+ m n− 1。证明了smnN（G）= m+ 2 mn − 2，并给出了η（G），s（G）的上界.此外，如果G是非循环幂零群，且p是G的最小素因子，|G|，我们证明了β（G）≤| G| p+ p− 1除非p= 2且G是双循环群，此时β（G）= 1 2| G| +2。
Let G be a multiplicative finite group and S= a 1⋅…⋅ a k a sequence over G. We call S a product-one sequence if 1=∏ i= 1 k a τ (i) holds for some permutation τ of {1,…, k}. The small Davenport constant d (G) is the maximal length of a product-one free sequence over G. For a subset L⊂ N, let s L (G) denote the smallest l∈ N 0∪{∞} such that every sequence S over G of length| S|≥ l has a product-one subsequence T of length| T|∈ L. Denote e (G)= max{ord (g): g∈ G}. Some classical product-one (zero-sum) invariants including D (G):= s N (G)(when G is abelian), E (G):= s {| G|}(G), s (G):= s {e (G)}(G), η (G):= s [1, e (G)](G) and s d N (G)(d∈ N) have received a lot of studies. The Noether number β (G) which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let G≅ C m⋉ φ C m n, in this paper, we prove that E (G)= d (G)+| G|= m 2 n+ m+ m n− 2 and β (G)= d (G)+ 1= m+ m n− 1. We also prove that s m n N (G)= m+ 2 m n− 2 and provide the upper bounds of η (G), s (G). Moreover, if G is a non-cyclic nilpotent group and p is the smallest prime divisor of| G|, we prove that β (G)≤| G| p+ p− 1 except if p= 2 and G is a dicyclic group, in which case β (G)= 1 2| G|+ 2.