I have come across the following interesting problem: let $p,q$ be polynomials over the field of real numbers, and let us suppose that their coefficients are all integer (that is, there is a finite exact representation of these polynomials). If needed, we may suppose that the degree of both polynomials is equal. Let us denote by $x_p$ (resp. $x_q$) the greatest absolute value of some (real or complex) root of the polynomial $p$ (resp. $q$). Is the property $x_p = x_q$ decidable?
If not, does this property hold for some restricted families of polynomials? In the context from which this problem arises, the polynomials are characteristic polynomials of matrices, and their roots are eigenvalues.
I am aware of some numerical algorithms for computing roots of polynomials / eigenvalues, however these seem to be of no use here, since the output of these algorithms is only approximate. It seems to me that computer algebra might be useful here, however, unfortunately, I do not have almost any knowledge in that field.
I am not searching for a detailed solution to this problem, however any intuition and ideas where to search for the solution would be helpful.
Thank you in advance.