On The Totally Real Algebraic Integers With Diameter Less Than 4

Let a be a totally real algebraic integer of degree d with minimal polynomial P(x)=xd+b1xd-1+…+bd-1x+dd,and α1=α,α2,…,αd be its all conjugates in a real interval[a,b].The diameter of a is diam(α)=(?)|αi-αj|In 1857,Kronecker[21]proved that there are infinitely many numbers a with b-a=4.In 1918,Schur and Polya[34]proved that there is only a finite number of algebraic integers with b-a<4.Then for b-a>4,Robinson[31]proved there are infinitely many numbers having this property.Thus the following question naturally arise:what happens when we only suppose that diam(α)<4?Until now people have computed all diameters less than 4 of degree up to 15.But the computing times increase too much with the increase of d,finding all the totally real algebraic integers with diameter less than 4 of a higher degree become more and more difficult.In this work,we use a type of auxiliary functions combined with Chebyshev polynomials,in order to improve the bands of sk,where sk=(?)αik.With this method,we get better bands of coefficients of P(x),and reduce computing times in finding the totally real algebraic integers with diameter less than 4.We then find all the totally real algebraic integers α with diam(α)<4 of degree d=16,17.