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Numerical Schubert Calculus

✍ Scribed by B. Huber; F. Sottile; B. Sturmfels

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
601 KB
Volume
26
Category
Article
ISSN
0747-7171

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✦ Synopsis

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a GrΓΆbner basis for the PlΓΌcker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.

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