2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67058We estabish an analog of the Cauchy-Poincare separation theorem for normal matrices in terms of majorization. Moreover, we present a solution to the inverse spectral problem (Borg-type result) for a normal matrix. Using this result we essentially generalize and complement the known Gauss--Lucas theorem on the geometry of the roots of a complex polynomial and of its derivative. In turn the last result is applied to prove the old conjectures of de Bruijn-Springer and Schoenberg about these roots.typos corrected; references addedComplex VariablesSpectral Theory15A29; 30C10; 30C15text