波利亚关于整系数幂级数猜想的思想研究

Pólya's Conjecture About Power Series with Integral Coefficients

整系数幂级数是级数理论研究中的一类重要级数,并有诸多成果问世。波利亚关于整系数幂级数猜想(后被称为波利亚-卡尔松定理)即是其中之一。文章基于原始文献,利用历史分析和比较的方法,探讨了波利亚提出该猜想的思想背景和影响。他在1915年提出该猜想,艾森斯坦、波莱尔、法都等人的工作为其重要基础;考察了波利亚在1921年、1928年、1931年关于该猜想的三篇文章,揭示了其思想的演变过程;最后探讨了波利亚的有关思想对其他数学家的一些重要影响。

Power series with integral coefficients are an important class in theoretical study of power series. Po1ya＇ s conjecture about power series with integral coefficients （ later called P61ya-Carlson theorem） is one of the many re- search results. This paper studies his four papers published in 1915,1921,1928 and 1931 and discusses its ideolog- ical background, the evolution and the influences of Po1ya＇ s conjecture. Po1ya proposed his conjecture in 1915 based on the work of G. Eisenstein, E. Borel, and P. Fatou.