Monic integer Chebyshev problem

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.