A problem in the real Schubert calculus

Jonathan Hauenstein and Frank Sottile
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We use alphaCertified to study one geometric problem from the point of view of some conjectures in the real Schubert calculus. General results in the Schubert calculus of enumerative geometry tell us that there will be exactly 126 four-dimensional linear subspaces of C8 that meet each of eight general three-dimensional linear subspaces nontrivially, in a one-dimensional linear subspace. The Shapiro Conjecture [S00] (now Theorem of Mukhin, Tarasov, and Varchenko[MTV]) asserts that if the three-planes are chosen in a specified way, then all solution planes will be real. The specified way is a follows. Let γ(t) be the (parametrized) moment curve in C8
γ(t) = (1, t, t1, t2, t3, t4, t5, t6, t7).
The three-plane F3(t) that osculates γ at a point γ(t) is the three-plane spanned by this vector γ(t), as well as its first and second derivatives. Then our three-planes are the osculating three-planes F3(ti) for eight real numbers t0, t1, t2, t3, t4, t5, t6, t7.
   We also tested closely related problems, where the eight numbers t0, …, t7 are not real, but are the roots of a polynomial with real coefficients. That is, some number a are real and the rest come in b complex conjugate pairs, where a+2b=8. Write this in short hand as arbc. The frequency table below presents the results of our investigation, which was intended to see if there was a lower bound on the number of real solutions, depending on the real scheme (arbc) of the points t0, …, t7. It does appear to be the case that there are non-trivial lower bounds for this Schubert problem. This is the largest problem that was investigated of this type (and the only one using numerical software), here is more on this question.
                      (W)8 = 126 on Gr(4;8)
type # real solutions
0246810 121416182022 242628303234 363840424446 485052545658 606264666870 727476788082 848688909294 9698100102104106 108110112114116118 120122124126 total
8r0c                                                                          1000 1000
6r1c    6610 8861683055418881832 174729162744416031873589 273223831846315014095135 1222917814675681697 93086820206716481023 236408734262380346 196162270155110416 100771746172103 6356421335250 18276692021 59000
4r2c      2614 377124971512328515791378 1074974567848523650 338268235377183239 1058972748364 695564464144 232331131420 7111431015 438656 321941 34138 24000
2r3c      8896 44791667162010797212586 595478215292188163 1066356622755 29161611153 476776 235112 211  1 21     1  1   1    23500
0r4c             19134 1585606287361158123 323528402820 7412281 624111 123211  1        1 1           1 23500

Before you begin, you will need to have a working binary of Bertini and alphaCertified on your machine along with the GMP library.
Assuming that we have the start points and created Shapiro.out, the following bash session computes and certifies the solutions for a random instance.
>: ./Shapiro.out
>: bertini Shapiro.bertini
>: alphaCertified Shapiro.poly nonsingular_solutions settings > output
>: sh scour.sh

[MTV]  E. Mukhin, V. Tarasov, A. Varchenko, The B  and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Annals of Mathematics, 170 (2009), 863—991.
[S00]  F. Sottile, Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro, Experimental Mathematics, 9 (2000), 161—182.
[S10]  F. Sottile, Frontiers of Reality in Schubert Calculus, Bulletin of the American Mathematical Society, 47 (2010), 31—71.

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Last modified: Wed Nov 3 18:28:32 CDT 2010