Eigenvalues of symmetric matrices over integral domains

Abstract

Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element. The latter result settles a questions raised by Bass, Estes and Guralnick.Comment: Serious error in previous versio

Characteristic polynomials of symmetric matrices over the univariate polynomial ring

“…For a slightly smaller class of polynomials, their result can be further extended: A monic polynomial over A is strictly real rooted if for any homomorphism A → R to a real closed field R all roots of the image of f in R[t] lie in R and are simple. Kummer recently showed in [Kum16] that for any integral domain A every strictly real rooted polynomial f ∈ A[t] divides the characteristic polynomial of a symmetric matrix.…”