Real Wronskian zero of polynomials with nonreal zeros
Let f be a polynomial with real coefficients and let \[ Wf(z)=f(z)f''(z)- (f'(z))^ 2 \] be the ``Wronskian'' of f. This paper is devoted to studying the relationship between the geometry of the zeros of f and that of Wf. The results are aimed at a conjecture of Craven, Csordas and Smith which states that the number of real zeros of Wf never exceeds that of f. The author's main results, as summarized in his introduction, are as follows (recall that the Jensen disk of a pair of conjugate complex numbers is the closed disk whose diameter is the line segment joining these two numbers): 1. Any nontrivial real zero of Wf lies in the Jensen disk of some pair of complex conjugate zeros of f. (A trivial real zero of Wf is one which is a multiple zero of f). 2. If a pair of complex conjugate zeros of f is sufficiently isolated from other zeros of f, then the associated closed Jensen disk contains exactly two real or complex zeros of Wf. 3. If there are real zeros of f sufficiently close to a pair of complex zeros of f, then the Jensen disk associated with these complex zeros contains no real zeros of Wf. 4. If, in a fixed interval, f has sufficiently many real zeros (depending on the number and location of the complex zeros of f), then Wf has no real zeros. 5. If the zeros of f lie in two circles symmetric to and sufficiently distant from the real axis, then Wf has exactly two real zeros. 6. If f has no multiple zeros, then the complex zeros of Wf outside the Jensen disks of f cannot lie too close to the real axis.
scientific article; zbMATH DE number 4177462