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) = ½. We make three general conjectures relating to the value of t M (I) for intervals I of length less than 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 functions L-(t) and L + (t), which 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.

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