A primal-dual formulation for certifiable computations in Schubert   calculus
  Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's \alpha-theory. 
Jonathan D. Hauenstein, Nickolas Hein, Frank Sottile
math.AG math.NA