Integer Symmetric Matrices: Counterexamples to Estes-Guralnick's conjecture

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 Estes--Guralnick'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 Schur--Siegel--Smyth trace problem for polynomials that appear as minimal polynomials of integer symmetric matrices or integer oscillatory matrices.