Monic integer Chebyshev problem

Abstract

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 $$t_M(E):=\underset{n\to \infty }{\lim }\Vert M_n{\Vert }_E^{1/n}.$$ 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 $a_1{b}_2-a_2{b}_1=1.$ 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.