Problems on the distribution of conjugates of algebraic numbers

Regular polygons and transfinite diameter

“…We can also study the links between diameters and characteristic values of a convex set (see also [1]), for example the t 3 diameter and the length of a convex set (see [6] and the definition of Section 2.1). Minimal diameters or weighted diameters of complete sets of conjugate algebraic integers for small degrees have been determined by the author and Lloyd-Smith [7,8,5]).…”

Diameters of complete sets of conjugate algebraic integers of small degree

“…Robinson [4] found real sets with diameters smaller than 4 and a cardinality smaller than 8, and has given a list for 7 and 8, without proving its completeness. Next, at the beginning of 1980, C. W. Lloyd-Smith [5] found these sets with diameters < 2 and a cardinality ≤ 5.…”

On a problem of Favard concerning algebraic integers

“…Favard [/], [2], Received 17 February 1984. I am grateful to one of the examiners of my thesis [5], who suggested the greatly improved technique for finding the ineguivalent algebraic integers with diameter less than some prescribed bound. I acknowledge the support of a Commonwealth Postgraduate Research Award while the research at the University of Adelaide was done.…”

“…The author [5] stated incorrectly that Favard found all inequivalent a with diam (a) < 2 when n = 2 or 3. However, this readily follows from the work of Favard [2].…”

“…The author [5] found all inequivalent a such that diam (a) < 2 when n = 4 or 5. The arguments are very complicated and yielded a total of 104,472 polynomials which had to be tested on the Cyber 173 computer at the University of Adelaide.…”