Consider Bethe ansatz equations for Gaudin model associated to gl(n).

Question. How many solutions are real, in the sense that polynomials with roots given by the solution have real coefficients?

This question is closely related to counting real points in intersections of Schubert varieties in Grassmannians of n-planes or to counting real Fuchsian differential operators of order n without monodromy and with prescribed non-negative integer ramification data. In the special case, the question is to count the preimages of real Wronski map.

Despite many efforts, the answer is not known in general. We will provide a lower bound based on a computation of signature of some natural Hermitian form.

This is a report on a joint work with V. Tarasov (IUPUI)