The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter t_M(I) is defined as the infimum of all such supremums. We show that if I has length 1 then t_M(I) = 1/2. We make three general conjectures relating to the value of t_M(I) for intervals I of length less that 4. We also conjecture a value for t_M([0, b]) where 0 < b < 1. We give some partial results, as well as computational evidence, to support these conjectures. We define two functions that measure properties of the lengths of intervals I with t_M(I) on either side of t. Upper and lower bounds are given for these functions. We also consider the problem of determining t_M(I) when I is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.Comment: 32 pages, 5 figure

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

Article electronically published on January 8, 2003 MONIC INTEGER CHEBYSHEV PROBLEM

Abstract. We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈Mn(Z) satisfies �Mn�E = inf Pn∈Mn(Z) �Pn�E. and the monic integer Chebyshev constant is then defined by tM (E): = lim n→ ∞ �Mn�1/n E. This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied. We compute tM (E) for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. Conjecture. Suppose [a2/b2,a1/b1] is an interval whose endpoints are consecutive Farey fractions. This is characterized by a1b2 − a2b1 =1. Then tM [a2/b2,a1/b1] =max(1/b1, 1/b2). 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. 1. Introduction an