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;C_n-1[t]) ---> P^k(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
type	1	3
2r1c	5685	4315
0r2c		10000

W) (W)⁶ = 9 on Gr(2;6)
type		# real solutions
		1	3	5	7	9
0c	4r1c		58233	16671	5495	19601
0c	2r2c	268	511	132	26	63
0c	0r3c		76009	18542	1248	4201

(W)⁸ = 14 on Gr(2;6)
	# real solutions
type	0	2	4	6	8	10	12	14
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
		1	3	5	7	9	11
2r0c	3r1c	226	691	474	136	67	406
2r0c	1r2c	901	593	297	82	35	92
0r1c	5r0c	347	597	396	117	91	452
0r1c	3r1c	475	1005	309	68	39	104
0r1c	1r2c		81859	12833	2039	1032	2237

X (X)⁶ = 16 on Gr(3;6)
type		# real solutions
		0	2	4	6	8	10	12	14	16
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
		1	3	5	7	9	11	13	15	17	19	21
1r	5r1c	79	427	1614	548	344	727	105	185	107	6	858
1r	3r2c	618	1708	1374	335	211	392	53	90	54	5	160
1r	1r3c		5754	2626	457	296	500	54	82	66	3	162

(X)⁹ = 42 on Gr(3;6)
	# real solutions
type	0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40	42
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
type	0	2	4	6
3r1c	247	289	185	279
1r2c		68044	21486	10470

(V)³ (V)⁴ = 13 on Gr(2;7)
type		# real solutions
		1	3	5	7	9	11	13
3r0c	2r1c			5087	1483	684	296	2450
3r0c	0r2c	17	23	45	8	3	2	2
1r1c	0r0c	31	13	4	14	10	3	25
1r1c	2r1c	638	391	444	292	113	41	81
1r1c	0r2c		6350	2353	890	195	93	119

(V)² (V)⁶ = 19 on Gr(2;7)
type		# real solutions
		1	3	5	7	9	11	13	15	17	19
2r0c	4r1c				56372	13010	6407	4790	1775	1222	16424
2r0c	2r2c		38101	18757	26416	7083	3123	2109	639	693	3079
2r0c	0r3c	26054	32176	18698	15245	3485	2113	637	215	464	913
0r1c	6r0c	352	88	46	80	60	56	72	51	17	178
0r1c	4r1c	347	125	110	158	99	75	26	13	12	35
0r1c	2r2c	289	231	110	205	79	47	17	11	2	9
0r1c	0r3c				7768	1529	528	73	39	22	41

V (V)⁸ = 28 on Gr(2;7)
	# real solutions
type	0	2	4	6	8	10	12	14	16	18	20	22	24	26	28
6r1c						12679	3370	1856	1283	754	454	420	229	169	3786
4r2c			7644	4693	2673	5103	1789	883	651	309	158	208	86	80	723
2r3c		5936	7947	4234	1971	2584	1030	454	369	122	66	61	18	43	165
0r4c			10216	6901	2051	3477	1153	392	493	94	42	37	15	44	85

(V)¹⁰ = 42 on Gr(2;7)
	# real solutions
type	0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40	42
8r1c								36255	12869	13072	6062	2456	5385	1408	1843	1142	2364	354	497	571	422	15300
6r2c				34152	12848	4772	8225	15552	6150	6351	2503	1533	2133	689	753	428	680	178	215	206	186	2446
4r3c		17431	18125	27288	9794	5390	5983	6075	3154	2431	1028	718	891	302	244	151	185	77	91	79	65	498
2r4c		32514	19740	20091	7270	5964	4686	3640	2521	1377	629	422	474	141	94	82	75	37	30	48	17	148

(W)⁴ = 8 on Gr(3;7)
# real solutions
type	0	2	4	6	8
2r1c	3590	292	6118
0r2c			10000

(W)⁶ = 16 on Gr(3;7)
	# real solutions
type	0	2	4	6	8	10	12	14	16
4r1c		4051	4395	9097	4959	2236	647	689	3926
2r2c			17397	6880	2693	1332	454	324	920
0r3c				7541	778	1427	31	70	153

(U)⁶ = 15 on Gr(2;8)
	# real solutions
type	1	3	5	7	9	11	13	15
4r1c		95926	32427	9639	12386	17390	5327	26905
2r2c		130489	28312	12541	11679	7507	2058	7414
0r3c				164540	19349	10548	3260	2303

(U)⁸ (U) = 20 on Gr(2;8)
type		# real solutions
		0	2	4	6	8	10	12	14	16	18	20
6r1c	1r						282976	39121	6899	5397	11200	54407
4r2c	1r			165393	73446	30332	94183	17094	3586	2767	3506	9693
2r3c	1r		99395	165903	64304	19189	36187	8255	1617	1291	1252	2607
0r4c	1r			247841	105047	10832	23264	9001	891	1111	584	1429

(U)⁶ (U)² = 20 on Gr(2;8)
type		# real solutions
		0	2	4	6	8	10	12	14	16	18	20
4r1c	2r0c					262715	31064	20542	16791	5271	4164	59453
2r2c	2r0c			198333	53393	103867	15161	8873	6383	1653	2225	10112
0r3c	2r0c	85534	59686	145049	42829	47329	8070	5381	1706	558	1216	2642
6r0c	0r1c	4464	922	519	219	645	378	364	517	355	106	1511
4r1c	0r1c	3766	1072	916	752	1438	798	564	240	144	75	235
2r2c	0r1c	2727	1471	1817	796	1940	667	343	102	54	25	58
0r3c	0r1c					196030	30047	10213	1709	805	476	720

(U)⁴ (U)⁴ = 30 on Gr(2;8)
type		# real solutions
		0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30
4r0c	2r1c			18204	7579	3290	3942	1629	955	1123	1730	1367	1412	1634	547	474	6114
2r1c	4r0c							28687	5488	3295	1871	973	873	424	492	385	7512
2r1c	2r1c		2179	2654	1306	696	791	1063	370	201	151	129	88	89	36	33	214
0r2c	4r0c				21676	7258	2636	11017	2645	1443	1063	194	274	192	194	139	1269
0r2c	2r1c			5070	2009	835	781	684	233	120	91	36	19	19	26	18	59

W W (W)⁴ = 6 on Gr(4;8)
type			# real solutions
			0	2	4	6
1r0c	1r0c	2r1c		79951		20049
1r0c	1r0c	0r2c		93154		6846

(W)⁴ = 9 on Gr(4;8)
type	# real solutions
	1	3	5	7	9
2r1c	4995	13	4692
0r2c			7500

W (W)⁷ = 20 on Gr(4;8)
type		# real solutions
		0	2	4	6	8	10	12	14	16	18	20
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
			0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30
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
		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
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
	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

Computations in Lagrangian Grassmannians

(X)² (X)² = 4 on L(3; 6)
type		# real solutions
		0	2	4
2r0c	2r0c	25000
2r0c	0r1c	6033	6972	11995
0r1c	2r0c	11906	7053	6041
0r1c	0r1c	25000

X (X)⁴ = 8 on L(3; 6)
type		# real solutions
		0	2	4	6	8
1r0c	4r0c	25000
1r0c	2r1c	11805	12215	980
1r0c	0r2c	11538	3220	6781	753	2708

(X)⁶ = 16 on L(3; 6)
	# real solutions
type	0	2	4	6	8	10	12	14	16
6r0c	25000
4r1c	25000
2r2c	172291	23874	194932	67798	135189	4741	175
0r3c	26229	4717	6312	2902	6889	395	714	122	1720

X (X)³ = 10 on L(4; 8)
type		# real solutions
		0	2	4	6	8	10
1r0c	3r0c	10000
1r0c	1r1c		14011	5989

X (X)² (X)² = 20 on L(4; 8)
type			# real solutions
			0	2	4	6	8	10	12	14	16	18	20
1r0c	2r0c	2r0c	19800
1r0c	2r0c	0r1c	5070	4494	5288	463	85
1r0c	0r1c	2r0c	7947	2169	387	97
1r0c	0r1c	0r1c			7478	810	912

(X)⁵ = 24 on L(4; 8)
type			# real solutions
	0	2	4	6	8	10	12	14	16	18	20	22	24
5r0c	500
3r1c	315	441	744	113	253	134
1r2c	500

Computations in Symplectic flag manifolds

(W)⁷ = 14 on L(2;6)
	# real solutions
type	0	2	4	6	8	10	12	14
7r0c	7517	7639	7807	16041	10960	36
5r1c		9307	15587	15115	9671	320
3r2c	6169	11901	10343	10286	10992	309
1r3c		21758	9796	6371	11878	197

Bibliography

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