Monic integer Chebyshev problem

• Main Authors: Borwein, P. B, Pinner, C. G, Pritsker, I. E
• Format: Journal Article
• Language: English
• Published: 19-07-2013
• Subjects: Mathematics - Number Theory
• DOI: 10.48550/arxiv.1307.5362

Math. Comp. 72 (2003), 1901-1916 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.