Diameters of complete sets of conjugate algebraic integers of small degree

Abstract

Abstract. We give bounds for the coefficients of a polynomial as functions of the diameter of its roots, hence we obtain polynomials with minimal diameters and small degree

Weighted diameters of complete sets of conjugate algebraic integers

“…BI = 0 PROOF: The bound Cd-n of t n (X d ) gives a bound for t2(Xd) thanks to formula (3.2) and the bounds given for the coefficients as a function of the diameter are proved in [4]. Furthermore [4] gives also good bounds for B2,B 3 ,B4 thanks to Newton's formulae, since we suppose the barycentre of the roots to be at the origin . Tables of i?2, -B3, B4 are given in Section 8.…”

“…The other results we know about t 2 » r e tQ e following: Favard [2] has computed inf t 2 (X), inf t 2 (X), and Lloyd-Smith [3] inf t 2 {X), inf t 2 {X). We have com- [4]. In [3], Lloyd-Smith also gives the following…”

“…D ALGORITHM. We fix Cdn and we find all polynomials with t n (Xd) < Cdn thanks to the algorithm given in [4]. We use formulae which 'translate' the barycentre at the origin:…”

Regular polygons and transfinite diameter

“…We remark that the sequence (t n (X)) is decreasing (see [7]). The transfinite diameter is also called the capacity, and is generally difficult to compute.…”

“…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]).…”