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.
Categories: math.NT