  Heine and Stieltjes in their studies of linear second-order differential
equations with polynomial coefficients having a polynomial solution of a
preassigned degree, discovered that the roots of such a solution are the
coordinates of a critical point of a certain remarkable symmetric function,
[He], [St]. Their result can be reformulated in terms of the Schubert calculus
as follows: the critical points label the elements of the intersection of
certain Schubert varieties in the Grassmannian of two-dimensional subspaces of
the space of complex polynomials, [S1].
  In a hundred years after the works of Heine and Stieltjes, it was established
that the same critical points determine the Bethe vectors in the sl_2 Gaudin
model, [G]. Recently it was proved that the Bethe vectors of the sl_2 Gaudin
model form a basis of the subspace of singular vectors of a given weight in the
tensor product of irreducible sl_2-representations, [SV].
  In the present work we generalize the result of Heine and Stieltjes to linear
differential equations of order p>2. The function, which determines elements in
the intersection of corresponding Schubert varieties in the Grassmannian of
p-dimensional subspaces, turns out to be the very function which appears in the
sl_p Gaudin model.
  In the case when the space of states of the Gaudin model is the tensor
product of symmetric powers of the standard sl_p-representation, we prove that
the Bethe vectors form a basis of the subspace of singular vectors of a given
weight.
