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数学进展
综述文章
C~n中有界域的Bergman核函数的零点问题
Zeroes of Bergman Kernel Function on Bounded Domains in C~n

殷慰萍;
YIN Weiping

首都师范大学数学系 北京,100037
(Department of Mathematics, Capital Normal University, Beijing, 100037, P. R. China

收稿日期: 2008-02-25
出版日期: 2008-02-25

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摘要 多复变数空间Cn中有界域的Bergman核函数的零点问题集中表现为陆启铿猜想.陆启铿猜想是波兰数学家M.Skwarczynski对陆启铿1966年的一篇文章中关于Bergman核函数的零点问题而命名的,至今已经40年了.该猜想已写入了多复变函数论的多本专著,引起很多数学家的兴趣而研究之,已经成为多复变函数论中的一个活跃的研究方向.本文简述了陆启铿猜想的最初含意,综述了迄今为止关于有界域的Bergman核函数有无零点的各种研究成果以及所用的思想和方法.特别对近来出现的陆启铿猜想的新研究领域进行了较详细的阐述并在最后提出了关于陆启铿猜想的6个Open Problems,希望国内的年轻数学家对陆启铿猜想感到兴趣而研究之.
关键词 陆启铿猜想陆启铿问题陆启铿域Bergman核函数的零点华罗庚域    
Abstract:The zeroes of Bergman kernel functions on bounded domains in Cn can be expressed by the Lu Qikeng conjecture mostly. In 1969 the Polish Mathematician M. Skwarczynski called the Lu Qikeng conjecture for the problem about zeroes of Bergman kernel function, that problem was appeared in the paper written by Lu Qikeng in 1966. Up to now it is 40 years over since the conjecture appeared. The conjecture was appeared in many monographes for several complex variables. Many mathematicians are interested in this conjecture and study it. Lu Qikeng conjecture is the active research direction in several complex variables. In the present paper, the motivation of Lu Qikeng is pointed out, the results, methods and ideas for the Lu Qikeng conjecture are summarized. The new research fileds of Lu Qikeng conjecture are discussed in detail. Six open problems are stated. The author hope the Lu Qikeng conjecture will be researched by the young scholars.
Key words Lu Qikeng problem    Lu Qikeng domain    zeroes of Bergman kernel function    Hua domain
[1] 赵晓霞,王安,张琳. 一类Reinhardt域的 Bergman核函数零点的边界性质[J]. 数学进展, 2014, 43(1): 64-68.
[2] 殷慰萍;. 华罗庚域研究的综述[J]. 数学进展, 2007, 36(2): 129-152.
[3] 殷慰萍. 有界域的Bergman核函数显式表示的最新进展[J]. 数学进展, 2002, 31(4): 295-312.
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