Frank Sottile
Texas A&M University
9 November 2005
Workshop on Real Algebra, Quadratic Forms, and Model Theory
Institut Henri Poincaré
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The Shapiro Conjecture Frank Sottile Texas A&M University 9 November 2005 Workshop on Real Algebra, Quadratic Forms, and Model Theory Institut Henri Poincaré |
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About 10 years ago, Boris Shapiro and Michael Shapiro
made a remarkable conjecture about real solutions to
geometric problems coming from the classical Schubert calculus.
While the conjecture remains open, there is truly overwhelming
computational evidence supporting it, and Eremenko and Gabrielov
proved it for Grassmannians of 2-planes, where the conjecture is
the appealing statement that a rational function with only real
critical points must be real.
In my talk, I will introduce the Shapiro conjecture and discuss what we know about it. This includes a simple counterexample and a refinement which is supported by massive experimental evidence. This evidence includes tantalizing computations which suggest a strengthening: that a certain discriminant polynomial is a sum of squares, or more generally that it has such an algebraic certificate of positivity. |
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