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- W2000517377 abstract "We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let $mathcal {M}_n({mathbb {Z}})$ denote the monic polynomials of degree $n$ with integer coefficients. A monic integer Chebyshev polynomial $M_n in mathcal {M}_n({mathbb {Z}})$ satisfies begin{equation*} | M_n |_{E} = inf _{P_n in mathcal {M}_n ( {mathbb {Z}})} | P_n |_{E}. end{equation*} and the monic integer Chebyshev constant is then defined by begin{equation*} t_M(E) := lim _{n rightarrow infty } | M_n |_{E}^{1/n}. end{equation*} This is the obvious analogue of the more usual 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. Conjecture. 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 begin{equation*}t_M[{a_2}/{b_2},{a_1}/{b_1}] = max (1/b_1,1/b_2).end{equation*} This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater." @default.
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- W2000517377 date "2003-01-08" @default.
- W2000517377 modified "2023-09-26" @default.
- W2000517377 title "Monic integer Chebyshev problem" @default.
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- W2000517377 doi "https://doi.org/10.1090/s0025-5718-03-01477-7" @default.
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