The monic integer transfinite diameter

Abstract:

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 $t_{\mathrm {M}}(I)$ is defined as the infimum of all such supremums. We show that if $I$ has length $1$, then $t_{\mathrm {M}}(I) = \tfrac {1}{2}$. We make three general conjectures relating to the value of $t_{\mathrm {M}}(I)$ for intervals $I$ of length less than $4$. We also conjecture a value for $t_{\mathrm {M}}([0,b])$ where $0<b\le 1$. 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 $t_{\mathrm {M}}(I)$ on either side of $t$. Upper and lower bounds are given for these functions. We also consider the problem of determining $t_{\mathrm {M}}(I)$ 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.