Beyond the Shapiro Conjecture and Eremenko-Gabrielov lower bounds

Jonathan Hauenstein, Nick Hein, Abraham Martín del Campo , and Frank Sottile.

Description   Notation   Data   Bibliography  


Description

   The Shapiro Conjecture for the Grassmannians (now Theorem of Mukhin, Tarasov, and Varchenko) posits that all the points in an intersection of Schubert varieties will be real, when those Schubert varieties are defined by flags osculating the rational normal curve at real points. This is implied via a standard limiting argument from the family of special cases in which the Schubert varieties all have codimension 1. This family of special cases is in turn is equivalent to the assertion that the Wronski map
(*)  
Wr :  Gr(k;Cn-1[t])  --->  Pk(n-k),
has only real points in its fibres over the set of hyperbolic polynomials, those polynomials with only real roots. This is a finite map from the Grassmannian of k-dimensional linear spaces of polynomials of degree n-1 to the projective space of polynomials of degree k(n-k),
   Eremenko and Gabrielov [EG] established the following weak form of the Shapiro Conjecture: They considered the real Wronski map, the restriction of (*) to the real points of the Grassmannian, which sends spaces of real polynomials to real Wronski polynomials. Their main result is that when n is odd, this map is surjective, and they computed a lower bound for its number of inverse images. (They also showed that when k=2, the map is not surjective when n is even.[])
   Here, we report on computations that go beyond these two results. We consider intersections of Schubert varieties given by flags osculating the rational normal curve where some of the flags are real and some come in complex conjugate pairs. (To ensure that the resulting intersection is defined over the real numbers R, we need that the two flags in each complex conjugate pair define Schubert varieties of the same type.)

Notation

   We use the following notation for Schubert conditions, which is uniform for the Frontiers of Reality project:
   The Schubert problem on Gr(2;6) denoted by (W)4 = 3 asks for the set of 2-planes in C6 that meet each of four given 3-planes. Here, the symbol W and its color indicate a codimension four linear subspace (W is fourth from the last in the Roman alphabet). The subscript is the Young diagram of the partition corresponding to the Schubert condition. The exponent 4 means that this Schubert condition which has codimension two, the number of boxes in the partition. W is repeated four times, and the = 3 indicates that there are three solutions to this problem. Observe that 2+2+2+2 is equal to the dimension of the Grassmannian Gr(2;6).
   There are three ways that this problem can be evaluated at flags osculating the rational normal curve, so that the resulting intersection is defined over R. Namely, evaluating the conditions at four real points, two real points and one complex conjugate pair, and at no real points and two complex conjugate pairs. We indicate these types by the shorthand 4r0c, 2r1c, and 0r2c. The first case of 4r0c is the Shapiro Conjecture (Theorem of Muknin, Tarasov, and Varchenko), and we do not test cases corresponding to the Shapiro Conjecture.

Data

Here are data from a small experiment that we conducted in October 2010 for several different Schubert problems on several different Grassmannians. We report these data in frequency tables. The columns represent the numbers of real solutions, and the rows represent the types of evaluation (as in 2r1c and 0r2c).
(W)4 = 3     on Gr(2;6)
# real solutions
type13
2r1c 56854315
0r2c  10000
W) (W)6 = 9     on Gr(2;6)
type # real solutions
135 79
0c 4r1c  5823316671 549519601
0c 2r2c 268511 13226 63
0c 0r3c  7600918542 12484201
(W)8 = 14     on Gr(2;6)
# real solutions
type02 468 101214
6r1c    3501 1643 1460 673 199 2524
4r2c  31317 27221 20417 11343 3407 1121 5174
2r3c 950 3692 1790 2574 650 153 45 146
0r4c     88180 9511 1207 290 812
(X)2 (X)5 = 11 on Gr(3;6)
type # real solutions
13 579 11
2r0c 3r1c 226691474 13667406
2r0c 1r2c 901593297 823592
0r1c 5r0c 347597396 11791452
0r1c 3r1c 4751005309 6839104
0r1c 1r2c  8185912833 203910322237
X (X)6 = 16 on Gr(3;6)
type # real solutions
024 6810 121416
1r 4r1c 21255 35482  17176  7228 18859
1r 2r2c 46357 32374  12268  3654 5347
1r 0r3c 60437 15351  18321  2116 3775
X (X)7 = 21 on Gr(3;6)
type # real solutions
13 579 111315 171921
1r 5r1c 794271614 548344727 105185107 6858
1r 3r2c 61817081374 335211392 539054 5160
1r 1r3c  57542626 457296500 548266 3162
(X)9 = 42 on Gr(3;6)
# real solutions
type 02 468 101214 161820 222426 283032 343638 4042
7r1c  1099  7975 42235  9081  6102 8827  1597  4207 1343  172  17362
5r2c  24495  30089 25992  5054  3632 4114  955  1586 832  63  3188
3r3c  39371  35022  15924  3150  1990  2183  494  622  367  35  842
1r4c     76117  14481  3574  1375  2925  271  364  204  32  477
(V)5 = 6     on Gr(2;7)
# real solutions
type02 46
3r1c 247289 185279
1r2c   68044 2148610470
(V)3 (V)4 = 13     on Gr(2;7)
type # real solutions
13 579 1113
3r0c 2r1c   5087 1483684296 2450
3r0c 0r2c 172345 832 2
1r1c 0r0c 31134 14103 25
1r1c 2r1c 638391444 29211341 81
1r1c 0r2c  63502353 89019593 119
(V)2 (V)6 = 19     on Gr(2;7)
type # real solutions
13 579 111315 1719
2r0c 4r1c     56372130106407 479017751222 16424
2r0c 2r2c  3810118757 2641670833123 2109639693 3079
2r0c 0r3c 260543217618698 1524534852113 637215464 913
0r1c 6r0c 3528846 806056 725117 178
0r1c 4r1c 347125110 1589975 261312 35
0r1c 2r2c 289231110 2057947 17112 9
0r1c 0r3c     77681529528 733922 41
V (V)8 = 28     on Gr(2;7)
# real solutions
type02 468 101214 161820 222426 28
6r1c       12679 337018561283 754454420 2291693786
4r2c   7644 469326735103 1789883651 309158208 8680723
2r3c  59367947 423419712584 1030454369 1226661 1843165
0r4c   10216 690120513477 1153392493 944237 154485
(V)10 = 42     on Gr(2;7)
# real solutions
type02 468 101214 161820 222426 283032 343638 4042
8r1c          3625512869 1307260622456 538514081843 11422364354 497571422 15300
6r2c     34152128484772 8225155526150 635125031533 2133689753 428680178 215206186 2446
4r3c  1743118125 2728897945390 598360753154 24311028718 891302244 15118577 917965 498
2r4c  3251419740 2009172705964 468636402521 1377629422 47414194 827537 304817 148
(W)4 = 8     on Gr(3;7)
# real solutions
type02 468
2r1c 35902926118   
0r2c   10000   
(W)6 = 16     on Gr(3;7)
# real solutions
type02 468 101214 16
4r1c  40514395 909749592236 6476893926
2r2c   17397 688026931332 454324920
0r3c     7541 7781427 3170 153
(U)6 = 15 on Gr(2;8)
# real solutions
type13 579 111315
4r1c  95926 324279639 1238617390 532726905
2r2c  130489 2831212541 116797507 20587414
0r3c     1645401934910548 32602303
(U)8 (U) = 20     on Gr(2;8)
type # real solutions
02 468 101214 161820
6r1c 1r        28297639121 68995397 1120054407
4r2c 1r    16539373446 3033294183 170943586 276735069693
2r3c 1r  99395 16590364304 1918936187 82551617 12911252 2607
0r4c 1r    247841105047 1083223264 9001891 1111584 1429
(U)6 (U)2 = 20 on Gr(2;8)
type # real solutions
02 468 101214 161820
4r1c2r0c       26271531064 2054216791 52714164 59453
2r2c2r0c    198333 53393 103867 15161 8873 6383 1653 2225 10112
0r3c2r0c 8553459686 14504942829 473298070 53811706 5581216 2642
6r0c0r1c 4464922 519219 645378 364517 355106 1511
4r1c 0r1c 37661072 916752 1438798 564240 14475 235
2r2c 0r1c 27271471 1817796 1940667 343102 5425 58
0r3c 0r1c      19603030047 102131709805 476720
(U)4 (U)4 = 30 on Gr(2;8)
type # real solutions
02 468 101214 161820 222426 2830
4r0c 2r1c   18204 757932903942 16299551123 173013671412 1634547474 6114
2r1c 4r0c         2868754883295 1871973873 424492385 7512
2r1c 2r1c  21792654 1306696791 1063370201 15112988 893633 214
0r2c 4r0c     2167672582636 1101726451443 1063194274 192194139 1269
0r2c 2r1c   5070 2009835781 684233120 913619 192618 59
W W (W)4 = 6 on Gr(4;8)
type # real solutions
02 46
1r0c 1r0c 2r1c  79951  20049
1r0c 1r0c 0r2c  93154  6846
(W)4 = 9 on Gr(4;8)
type # real solutions
13 579
2r1c 4995134692   
0r2c   7500   
W (W)7 = 20 on Gr(4;8)
type # real solutions
02 468 101214 161820
1r0c 5r1c      90088       19912
1r0c 3r2c   64725  40087       5188
1r0c 1r3c 27041 58900  21784       2275
W W (W)5 = 30 on Gr(4;8)
type # real solutions
02 468 101214 161820 222426 2830
1r0c 1r0c 3r1c     49016 17671  8986  5890 1052  2304  15081
1r0c 1r0c 1r2c  42741  35629 9520  4283  3028 744  828  3227
                      W (W)8 = 90 on Gr(4;8)
type # real solutions
02 468 101214 161820 222426 283032 343638 404244 464850 525456 586062 646668 707274 767880 828486 8890
1r0c 6r1c             1586 510  270  924 118  86  348 44  39  70 123  15  36 16  8  100 1  1  705
1r0c 4r2c     1611 850  423  557 373  195  432 42  61  179 21  23  31 26  5  19 3  1  46       102
1r0c 2r3c  1080  1708 773  394  319 210  85  198 25  23  84 11  9  14 10  2  14    1  15       25
1r0c 0r4c     2756 807  747  162 178  59  95 23  34  89    4  6 5     12       5 1     17
                      (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


Computations in Lagrangian Grassmannians


(X)2 (X)2 = 4 on L(3; 6)
type # real solutions
024
2r0c 2r0c 25000  
2r0c 0r1c 6033697211995
0r1c 2r0c 1190670536041
0r1c 0r1c 25000  
X (X)4 = 8 on L(3; 6)
type # real solutions
02 468
1r0c 4r0c 25000     
1r0c 2r1c 1180512215980   
1r0c 0r2c 1153832206781 7532708
(X)6 = 16 on L(3; 6)
  # real solutions
type 02 468 101214 16
6r0c 25000          
4r1c 25000          
2r2c 17229123874194932 677981351894741 175  
0r3c 2622947176312 29026889395 7141221720
X (X)3 = 10 on L(4; 8)
type # real solutions
024 6810
1r0c 3r0c 10000      
1r0c 1r1c  140115989    
X (X)2 (X)2 = 20 on L(4; 8)
type # real solutions
024 6810 1214 161820
1r0c 2r0c 2r0c 19800             
1r0c 2r0c 0r1c 507044945288 46385        
1r0c 0r1c 2r0c 79472169387 97         
1r0c 0r1c 0r1c   7478 810912        
(X)5 = 24 on L(4; 8)
type # real solutions
024 6810 1214 161820 2224
5r0c 500                
3r1c 315441744 113253134          
1r2c 500                

Computations in Symplectic flag manifolds


(W)7 = 14     on L(2;6)
# real solutions
type02 468 101214
7r0c 751776397807 160411096036   
5r1c  930715587 151159671320   
3r2c 61691190110343 1028610992309   
1r3c  217589796 637111878197   

Bibliography


Last modified: Fri Mar 25 16:12:57 CET 2011