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Tropical geometry in polynomial system solving

Anders Nedergaard Jensen
Algebraseminar
Torsdag, 10 december, 2015, at 15:15-16:15, in Aud. D3 (1531-215)
Abstrakt:
Tropical geometry is a piecewise linear version of algebraic geometry, where polyhedral complexes play the role of algebraic sets. In this talk we will see several examples of how that makes tropical geometry useful for polynomial system solving. Our main example will be an attempt to tropicalise numerical algebraic geometry, where polynomial equations are solved via homotopy continuation. Regular triangulations of point configurations will play an important role as the homotopy process is discretised. The outcome is an algorithm for computing the mixed volume and mixed cells of a square polynomial system, which in turn allows us to apply the Huber-Sturmfels numerical homotopy realisation of Bernstein's bound on the number of isolated solutions. We end by mentioning two other projects where tropical geometry has applications in polynomial system solving. One is related to Smale's 6th problem in celestial mechanics (with Marshall Hampton) and the other is on tropical resultants (with Josephine Yu).
Kontaktperson: Anders Nedergaard jensen