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
Key words
integer Chebyshev problem/integer transfinite diameter/integer Chebyshev polynomial/critical polynomial/necessary factor/auxiliary function
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