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(t_i) 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
	0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40	42	44	46	48	50	52	54	56	58	60	62	64	66	68	70	72	74	76	78	80	82	84	86	88	90	92	94	96	98	100	102	104	106	108	110	112	114	116	118	120	122	124	126	total
8r0c																																																																1000	1000
6r1c				6	6	10	88	616	830	554	1888	1832	1747	2916	2744	4160	3187	3589	2732	2383	1846	3150	1409	5135	1222	917	814	675	681	697	930	868	2020	671	648	1023	236	408	734	262	380	346	196	162	270	155	110	416	100	77	174	61	72	103	63	56	42	133	52	50	182	76	69	2021	59000
4r2c						2614	3771	2497	1512	3285	1579	1378	1074	974	567	848	523	650	338	268	235	377	183	239	105	89	72	74	83	64	69	55	64	46	41	44	23	23	31	13	14	20	7	11	14	3	10	15	4	3	8	6	5	6	3	2	1	9	4	1	3	4	1	38	24000
2r3c						8896	4479	1667	1620	1079	721	2586	595	478	215	292	188	163	106	63	56	62	27	55	29	16	16	11	15	3	4	7	6	7	7	6	2	3	5	1	1	2	2	1	1			1	2	1					1			1			1				23500
0r4c												19134	1585	606	287	361	158	123	32	35	28	40	28	20	7	4	12	2	8	1	6	2	4	1	1	1	1	2	3	2	1	1		1								1		1										1	23500

Directions
Before you begin, you will need to have a working binary of Bertini and alphaCertified on your machine along with the GMP library.

Solve for a random complex instance by running Bertini with input file ShapiroOrig.bertini.
>: bertini ShapiroOrig.bertini
Using a single processor, this computation takes many hours.
The output file of interest is nonsingular_solutions, which will be the start points for the parameter homotopy.
Rename nonsingular_solutions as start. These will be the start points for the parameter homotopy used to solve the real instances.
>: mv nonsingular_solutions start
Set the number of real points to utilize in ShapiroInstance.c++ by setting the preprocessor directive NUMREAL to the number of real points to utilize (that is, the a in arbc).
Create the executable Shapiro.out by compiling ShapiroInstance.c++ using the header file Complex.h. This executable creates the input files needed to solve for a random instance.
>: g++ -o Shapiro.out ShapiroInstance.c++ -lgmp
Run Shapiro.out. This creates Shapiro.bertini, which is a parameter homotopy to compute numerical solutions for a random instance, and Shapiro.poly, which is the polynomial system used to certify the solutions. For example, Shapiro.bertini and Shapiro.poly.
>: ./Shapiro.out
Use the parameter homotopy described in Shapiro.bertini to compute the numerical solutions.
>: bertini Shapiro.bertini
The output file of interest is nonsingular_solutions, which is part of the input for alphaCertified. Here is nonsingular_solutions for the running example.
We now use alphaCertified to certify the number of real solutions. This is accomplished by running alphaCertified with the polynomial system described in Shapiro.poly using the points contained in nonsingular_solutions created by Bertini in the last step. The configurations in settings tell alphaCertified to utilize 256-bit precision. We recommend capturing the output for perusal.
>: alphaCertified Shapiro.poly nonsingular_solutions settings > output
Here is output for the running example. Perusing it, we see that alphaCertified certified that 16 of the solutions are real.
Bertini and alphaCertified both create a halo of small files, which may be removed using the short shell script scour.sh.
>: sh scour.sh

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