4,128 research outputs found
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 Gr\"obner basis for the
Pl\"ucker 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.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st
Galois groups of Schubert problems via homotopy computation
Numerical homotopy continuation of solutions to polynomial equations is the
foundation for numerical algebraic geometry, whose development has been driven
by applications of mathematics. We use numerical homotopy continuation to
investigate the problem in pure mathematics of determining Galois groups in the
Schubert calculus. For example, we show by direct computation that the Galois
group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes
non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde
A primal-dual formulation for certifiable computations in Schubert calculus
Formulating a Schubert problem as the solutions to a system of equations in
either Pl\"ucker space or in the local coordinates of a Schubert cell typically
involves more equations than variables. We present a novel primal-dual
formulation of any Schubert problem on a Grassmannian or flag manifold as a
system of bilinear equations with the same number of equations as variables.
This formulation enables numerical computations in the Schubert calculus to be
certified using algorithms based on Smale's \alpha-theory.Comment: 21 page
Numerical Schubert Calculus by the Pieri Homotopy Algorithm
[[abstract]]Based on Pieri's formula on Schubert varieties, the Pieri homotopy algorithm was first proposed by Huber, Sottile, and Sturmfels [J. Symbolic Comput., 26 (1998), pp. 767-788] for numerical Schubert calculus to enumerate all p-planes in Cm+p that meet n given planes in general position. The algorithm has been improved by Huber and Verschelde [SIAM J. Control Optim., 38 (2000), pp. 1265-1287] to be more intuitive and more suitable for computer implementations. A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm.[[notice]]補正完畢[[incitationindex]]SCI[[booktype]]紙本[[booktype]]電子
Some real and unreal enumerative geometry for flag manifolds
We present a general method for constructing real solutions to some problems
in enumerative geometry which gives lower bounds on the maximum number of real
solutions. We apply this method to show that two new classes of enumerative
geometric problems on flag manifolds may have all their solutions be real and
modify this method to show that another class may have no real solutions, which
is a new phenomenon. This method originated in a numerical homotopy
continuation algorithm adapted to the special Schubert calculus on
Grassmannians and in principle gives optimal numerical homotopy algorithms for
finding explicit solutions to these other enumerative problems.Comment: 19 pages, LaTeX-2e; Updated and final version. To appear in the issue
of Michigan Mathematical Journal dedicated to Bill Fulto
The Secant Conjecture in the real Schubert calculus
We formulate the Secant Conjecture, which is a generalization of the Shapiro
Conjecture for Grassmannians. It asserts that an intersection of Schubert
varieties in a Grassmannian is transverse with all points real, if the flags
defining the Schubert varieties are secant along disjoint intervals of a
rational normal curve. We present theoretical evidence for it as well as
computational evidence obtained in over one terahertz-year of computing, and we
discuss some phenomena we observed in our data.Comment: 19 page
The special Schubert calculus is real
We show that the Schubert calculus of enumerative geometry is real, for
special Schubert conditions. That is, for any such enumerative problem, there
exist real conditions for which all the a priori complex solutions are real.Comment: 5 page
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