The EGZ-constant and short zero-sum sequences over finite abelian groups

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• 作者：Weidong Gao ; ^{wdgao1963@aliyun.com" class="auth_mail" title="E-mail the corresponding author} ; Dongchun Han ^{han-qingfeng@163.com" class="auth_mail" title="E-mail the corresponding author} ; Hanbin Zhang ^{nkuzhanghanbin@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Zero-sum sequence ; Short zero-sum sequence ; Davenport constant ; EGZ-constant
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：162
• 期：Complete
• 页码：601-613
• 全文大小：308 K

文摘

Let G   be an additive finite abelian group with exponent pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si1.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=ae3fec66ea8e3cd3e3de4244bea5a3ed" title="Click to view the MathML source">exp⁡(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\exp (G)$pan>pan>pan>. Let pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si2.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=5a8fcf810daab4f87843c147ebc0b651" title="Click to view the MathML source">η(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\eta (G)$pan>pan>pan> be the smallest integer t such that every sequence of length t   has a nonempty zero-sum subsequence of length at most pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si1.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=ae3fec66ea8e3cd3e3de4244bea5a3ed" title="Click to view the MathML source">exp⁡(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\exp (G)$pan>pan>pan>. Let pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si3.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=d0193b270d6dfd382cda803d919c9f0e" title="Click to view the MathML source">s(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathsf{s}(G)$pan>pan>pan> be the EGZ-constant of G, which is defined as the smallest integer t such that every sequence of length t   has a zero-sum subsequence of length pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si1.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=ae3fec66ea8e3cd3e3de4244bea5a3ed" title="Click to view the MathML source">exp⁡(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\exp (G)$pan>pan>pan>. Let p   be an odd prime. We determine pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si2.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=5a8fcf810daab4f87843c147ebc0b651" title="Click to view the MathML source">η(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\eta (G)$pan>pan>pan> for some groups G   with pan id="mmlsi27" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si27.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=991577ffd7420f84afd40308c0034452" title="Click to view the MathML source">D(G)≤2exp⁡(G)−1pan>pan class="mathContainer hidden">pan class="mathCode">$\mathsf{D}(G)\le 2\exp (G)-1$pan>pan>pan>, including the p-groups of rank three and the p  -groups pan id="mmlsi5" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$G=C_{\exp (G)}\oplus C_{p^m}^r$pan>pan>pan>. We also determine pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si3.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=d0193b270d6dfd382cda803d919c9f0e" title="Click to view the MathML source">s(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathsf{s}(G)$pan>pan>pan> for the groups G   above with more larger exponent than pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si7.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=585ad9a6d4945fcca21f1ba45859923c" title="Click to view the MathML source">D(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathsf{D}(G)$pan>pan>pan>, which confirms a conjecture by Schmid and Zhuang from 2010, where pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003327&_mathId=si7.gif&_user=111111111&_pii=S0022314X15003327&_rdoc=1&_issn=0022314X&md5=585ad9a6d4945fcca21f1ba45859923c" title="Click to view the MathML source">D(G)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathsf{D}(G)$pan>pan>pan> denotes the Davenport constant of G.

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