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3. General Schubert conditions and overdetermined systems

Previous sections dealt with simple hypersurface Schubert conditions. Those had intimate connections with the pole placement problem from systems theory and led to interesting systems of polynomials. The conjecture of Shapiro and Shapiro can be made for general Schubert conditions on a Grassmannian, which we describe here. There is ample computational evidence and some proofs in support of this more general version of their Conjecture.
  1. The Schubert calculus for the Grassmannian.
  2. The conjecture of Shapiro and Shapiro.
  3. Local coordinates.
  4. Equations.
  5. Proof in some cases.
  6. Computational evidence.
  7. Challenge problems.
4. Total positivity.
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