Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let \({\M}_n({\Z})\) denote the monic polynomials of degree \(n\) with integer coefficients. A {\it monic integer Chebyshev polynomial} \(M_n \in {\M}_n({\Z})\) satisfies $$ \| M_n \|_{E} = \inf_{P_n...

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Bibliographic Details
Published inarXiv.org
Main Authors Borwein, P B, Pinner, C G, Pritsker, I E
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 20.07.2013
Subjects
Chebyshev approximation
Intervals
Polynomials
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Summary:We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let \({\M}_n({\Z})\) denote the monic polynomials of degree \(n\) with integer coefficients. A {\it monic integer Chebyshev polynomial} \(M_n \in {\M}_n({\Z})\) satisfies $$ \| M_n \|_{E} = \inf_{P_n \in{\M}_n ({\Z})} \| P_n \|_{E}. $$ and the {\it monic integer Chebyshev constant} is then defined by $$ t_M(E) := \lim_{n \rightarrow \infty} \| M_n \|_{E}^{1/n}. $$ This is the obvious analogue of the more usual {\it integer Chebyshev constant} that has been much studied. We compute \(t_M(E)\) for various sets including all finite sets of rationals and make the following conjecture, which we prove in many cases. \medskip\noindent {\bf Conjecture.} {\it Suppose \([{a_2}/{b_2},{a_1}/{b_1}]\) is an interval whose endpoints are consecutive Farey fractions. This is characterized by \(a_1b_2-a_2b_1=1.\) Then} $$t_M[{a_2}/{b_2},{a_1}/{b_1}] = \max(1/b_1,1/b_2).$$ This should be contrasted with the non-monic integer Chebyshev constant case where the only intervals where the constant is exactly computed are intervals of length 4 or greater.
ISSN:2331-8422