整Chebyshev问题及其应用  

The integer Chebyshev problem and its applications

作　　者：王聪[1,2] 吴强 Cong Wang;Qiang Wu

机构地区：[1]西南大学数学与统计学院,重庆400715 [2]重庆医科大学医学信息学院,重庆400016

出　　处：《中国科学：数学》2024年第9期1365-1390,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12071375)资助项目。

摘　　要：整Chebyshev问题是要寻找一个次数不超过n的非零整系数多项式,使其在给定区间上的绝对值的最大值(即上确界范数)最小,并分析当n趋于无穷时,该最小上确界范数的变化趋势.本文概述了该问题的研究历史、方法及其推广和应用,并讨论了两类长度小于4的区间上的整Chebyshev问题.我们发现上述区间上具有最小上确界范数的n(当n足够大时)次整系数多项式的部分因子都具有一种特定的性质,并猜测该性质对任意同类区间都成立.The integer Chebyshev problem aims to find a nonzero polynomial of degree at most n,with integer coefficients,that has the smallest possible supremum norm on a given interval,and to analyze the behavior as n tends to infinity.In this paper,we summarize the research history and research methods of this problem and introduce its generalization and applications.Then we study this problem on two types of intervals with lengths less than 4.We find that for the polynomials with integer coefficients of degree n(when n is large enough),having small supremum norm on the above intervals,some of their factors have a certain property.We conjecture that this property holds for all intervals of these two types.

关 键 词：整Chebyshev问题 整超限直径 整Chebyshev多项式 关键多项式 必需因子 辅助函数 

分 类 号：O174.14[理学—数学]