2. Complete intersections: Hypersurface Schubert conditions
In its simplest form, the conjecture of Shapiro and Shapiro asserts that
given mp general m-planes in (m+p)-space which osculate
a real rational normal curve, each of the (finitely many) p-planes
meeting all mp is real.
This is a special case of the real pole placement problem.
Using local coordinates for the Grassmannian, we may represent these
geometric problems as systems of polynomial equations.
This formulation is amenable to study with symbolic and numeric methods, and
we can verify and prove numerous cases of the conjecture, which we document
here.
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Polynomial formulation of hypersurface Schubert conditions.
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The conjecture of Shapiro and Shapiro.
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The pole placement problem and geometry.
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Shapiro's conjecture and the placement problem.
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Equivalent systems of polynomials.
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Proof when (m,p)=(2,3).
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Computational evidence.
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Complexity of these computations.
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Why these equations are interesting.
3.
General Schubert conditions and overdetermined systems.