The monic integer transfinite diameter

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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 tₘ(I) is defined as the infimum of all such supremums. We show that if I has length 1 then tₘ(I) = 1/2. We make three general conjectures relating to the value of tₘ(I) for intervals I of length less that 4. We also conjecture a value for tₘ([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 tₘ(I) on either side of t. Upper and lower bounds are given for these functions. We also consider the problem of determining tₘ(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.