Numerical Schubert Calculus
Copyright policy )Numerical Schubert Calculus
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 Gröbner basis for the Plücker 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.
24 pages, LaTeX 2e with 2 figures, used epsf.sty
- University of Toronto Canada
- The University of Texas System United States
- Mathematical Sciences Research Institute United States
- University of Wisconsin–Oshkosh United States
- Texas A&M Research Foundation United States
Configurations and arrangements of linear subspaces, Numerical Analysis, systems of polynomial equations, Algebra and Number Theory, Schubert calculus, Primary 65H10 Secondary 14N10, 14M15, SAGBI basis, Numerical Analysis (math.NA), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), numerical homotopy algorithms, Symbolic computation and algebraic computation, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Computational Mathematics, Enumerative problems (combinatorial problems) in algebraic geometry, Computational aspects of higher-dimensional varieties, Optimization and Control (math.OC), Optimization and Control, FOS: Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)
Configurations and arrangements of linear subspaces, Numerical Analysis, systems of polynomial equations, Algebra and Number Theory, Schubert calculus, Primary 65H10 Secondary 14N10, 14M15, SAGBI basis, Numerical Analysis (math.NA), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), numerical homotopy algorithms, Symbolic computation and algebraic computation, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Computational Mathematics, Enumerative problems (combinatorial problems) in algebraic geometry, Computational aspects of higher-dimensional varieties, Optimization and Control (math.OC), Optimization and Control, FOS: Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)
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