Three hard polynomial systems from enumerative geometry

I will  briefly discuss three problems  from enumerative geometry
and their  formulation as  systems of polynomials.   One involves
Galois groups  of Fano  problems, the  other a  possible relation
between Welschinger  signs and the  degree of the Wronski  map in
the real  Schubert calculus,  and the  third is  an interpolation
problem.  These  systems appear challenging (or  not possible) to
solve, which has impeded some recent investigations.