  We 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.
