Abstract
The aim of this thesis is to study which polynomials appear as minimal polynomials of integer symmetric matrices. It has been known for a long time that to be the minimal polynomial of a rational symmetric matrix it is necessary and sufficient that the polynomial is monic, separable and has only real roots. It was conjectured by Estes and Guralnick that the equivalent conditions should hold for integer symmetric matrices.
We present counterexamples to EstesGuralnick's conjecture for every degree strictly larger than five. In the process, we construct Salem numbers of trace $2$ for every even degree strictly larger than $22$. Furthermore, we settle the SchurSiegelSmyth trace problem for polynomials that appear as minimal polynomials of integer symmetric matrices or integer oscillatory matrices.
We present counterexamples to EstesGuralnick's conjecture for every degree strictly larger than five. In the process, we construct Salem numbers of trace $2$ for every even degree strictly larger than $22$. Furthermore, we settle the SchurSiegelSmyth trace problem for polynomials that appear as minimal polynomials of integer symmetric matrices or integer oscillatory matrices.
Original language  English 

Qualification  Ph.D. 
Awarding Institution 

Supervisors/Advisors 

Award date  1 Apr 2017 
Publication status  Unpublished  2016 