Monic integer Chebyshev problem

TL;DR

This paper investigates the minimal supremum norm of monic polynomials with integer coefficients over sets, introduces the monic integer Chebyshev constant, computes it for various sets, and conjectures a formula for intervals with Farey fraction endpoints.

Contribution

It defines the monic integer Chebyshev constant, computes it for specific sets, and proposes a conjecture relating it to Farey fractions, extending the understanding of integer Chebyshev problems.

Findings

Computed $t_M(E)$ for various sets including rationals.

Formulated and proved a conjecture for intervals with Farey fraction endpoints.

Contrasted monic and non-monic integer Chebyshev constants.

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 $$ t_M(E) := \lim_{n \rightarrow \infty} \| M_n \|_{E}^{1/n}. $$ 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.