Waring问题的重要进展——迭代方法的改进  

The Important Advance on Waring′s Problem ——Improvement of Iterative Method

作　　者：陆鸣皋[1] 

机构地区：[1]上海大学数学系

出　　处：《数学进展》1998年第4期289-300,共12页Advances in Mathematics（China）

基　　金：国家自然科学基金

摘　　要：近些年来，Ｗａｒｉｎｇ问题取得了重要的进展．本文主要阐明了Ｗａｒｉｎｇ问题研究中迭代方法的发展历史以及最近的重要改进，给出了Ｇ（ｋ）（表示使得每个充分大的自然数都能表成至多是ｓ个正整数ｋ次方之和的最小ｓ）的最新上界的证明思路，使得具有大学数学专业水平的读者在阅读后对这个著名问题会有比较深入的了解．文章最后指出了Ｖａｕｇｈａｎ的ｐ－ａｄｉｃ迭代方法可应用到非齐次的Ｗａｒｉｎｇ问题（Ｗａｒｉｎｇ型问题）中去，从而得到一些新的深刻结果．In the past few year considerable progress has been made in regard to Waring′s problem. In this article we briefly describe the developments of iterative method in Waring′s problem and give the outline of proofs of Vaughan-Wooley′s upper bounds of G(k) , that is, the smallest s such that every sufficiently large natural number is the sum of at most s k -th powers of natural number. Finally, it is pointed out that Vaughan′s p -adic iterative method can be applied to the problem of sums of mixed powers (Waring Type Problem) and some comparable results have been obtained.

关 键 词：WARING问题 H-L法 迭代方法 混合幂和 

分 类 号：O156.4[理学—数学]