Perturbations on constructible lists preserving realizability in the NIEP and questions of Monov

In [5] the authors showed that if σ=(λ1,λ2,λ2¯,λ4,…,λn) is realizable where λ1λ1 is the Perron eigenvalue and λ2λ2 is non-real, then so too is σ=(λ1+4t,λ2+t,λ2¯+t,λ4,…,λn). They asked if 4 can be replaced by 1 or 2 or what is the least possible multiple c⩾1c⩾1 of t in order for this perturbation to be realizable. In [2] the authors showed that for n=4n=4 one can find certain spectra for which the result holds when c=1c=1 provided t   is “small”. In this paper we show that c=2c=2 is best possible and we construct a realizing matrix for c=2c=2 when t is sufficiently small. We also address some questions of Monov concerning the realizability of the derivative of a realizable polynomial and if such a polynomial must have positive power sums.