The Complexity of Generalized Center Problem

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• 作者：Guangfeng Dong (1)
  Changjian Liu (2)
  Jiazhong Yang (3)
  
  1. Department of Mathematics ; Sun Yat-Sen University ; Guangzhou ; 510275 ; People鈥檚 Republic of China
  2. School of Mathematics ; Soochow University ; Suzhou ; 215006 ; People鈥檚 Republic of China
  3. School of Mathematical Sciences ; Peking University ; Beijing ; 100871 ; People鈥檚 Republic of China
  
• 关键词：Polynomial differential systems ; $$p ; q$$ p ; ; q Resonance ; Saddle value ; Saddle order ; Generalized center ; 34C05 ; 34C20 ; 34M35 ; 37G05
• 刊名：Qualitative Theory of Dynamical Systems
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：14
• 期：1
• 页码：11-23
• 全文大小：219 KB
• 参考文献：1. Apostol, TM (1976) Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics. Springer Science & Business Media, New York
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  19. Qiu, Y, Yang, J (2009) On the focus order of planar polynomial differential equations. J. Differ. Equ. 246: pp. 3361-3379 CrossRef
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Dynamical Systems and Ergodic Theory
  Difference and Functional Equations
  
• 出版者：Birkh盲user Basel
• ISSN：1662-3592

文摘

In this paper, we study the complexity of generalized center problem by looking for the lower bounds of the maximal possible saddle order \(M(p,q,n)\) of planar polynomial differential equations of arbitrary degree \(n\) admitting any \(p:-q\) saddle type resonance between the eigenvalues. We prove that, for any given positive integers \(p\) and \(q\) , and sufficiently big integer \(n\) , \(M(p,q,n)\) can grow at least as rapidly as \(2n^2\) . This result significantly improves all the known lower bounds which are at most \(n^2+O(n)\) .