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The Conjecture of Shapiro and Shapiro
The Conjecture of Shapiro and Shapiro
Frank Sottile
sottile@math.tamu.edu
http://www.math.tamu.edu/~sottile/
18 May 2004
Table of Contents
Summary
 Background.
 Real enumerative geometry.
 Linear systems.
 Pole placement problem.
 History of the conjectures.
 Complete intersections: hypersurface Schubert
conditions.

Polynomial formulation of hypersurface Schubert conditions.

The conjecture of Shapiro and Shapiro.

The pole placement problem and geometry.

Shapiro's conjecture and the pole placement problem.

Equivalent systems of polynomials.

Proof when (m,p)=(2,3).

Computational evidence.

Complexity of these computations.

Why these equations are interesting.
 General Schubert conditions and overdetermined
systems.

The Schubert calculus for the Grassmannian.

The conjecture of Shapiro and Shapiro.

Local coordinates for the intersection of 2 Schubert varieties.

Equations for Schubert varieties.

Proof in some cases.

Computational evidence.

Challenge problems.
 Total positivity.

Total positivity and the conjecture of Shapiro and Shapiro.

Parameterization of totally positive matrices.

Computational evidence.
 Further remarks.

A counterexample to the original conjecture of Shapiro and Shapiro.

Further questions.

Recent developments.
 Acknowledgements.
 Bibliography.
Last Modified 17 April 2000.