The monotone secant conjecture in the real Schubert calculus
Nickolas Hein, Christopher J. Hillar, Abraham Martin del Campo, Frank Sottile, Zach Teitler
aa r X i v : . [ m a t h . AG ] O c t THE MONOTONE SECANT CONJECTURE IN THEREAL SCHUBERT CALCULUS
NICKOLAS HEIN, CHRISTOPHER J. HILLAR, ABRAHAM MART´IN DEL CAMPO,FRANK SOTTILE, AND ZACH TEITLER
Abstract.
The monotone secant conjecture posits a rich class of polynomial systems,all of whose solutions are real. These systems come from the Schubert calculus on flagmanifolds, and the monotone secant conjecture is a compelling generalization of the Shapiroconjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We presentsome theoretical evidence for this conjecture, as well as computational evidence obtainedby 1.9 teraHertz-years of computing, and we discuss some of the phenomena we observed inour data. Introduction
A system of real polynomial equations with finitely many solutions has some, but likelynot all, of its solutions real. In fact, sometimes the structure of the equations implies anupper bound on the number of real solutions [2, 12], ensuring that not all solutions are real.The monotone secant conjecture posits a family of systems of polynomial equations with theextreme property that all of their solutions are real.The Shapiro conjecture asserts that a zero-dimensional intersection of Schubert subvari-eties of a Grassmannian consists only of real points provided that the Schubert varieties aregiven by flags tangent to a real rational normal curve. While the statement concerns theSchubert calculus on Grassmannians, its proofs involve complex analysis [5, 6] or integrablesystems and representation theory [14, 15]. A complete story of this conjecture and its proofcan be found in [20].The Shapiro conjecture is false for non-Grassmannian flag manifolds, but in a very inter-esting manner. This failure was first noticed in [18] and systematic computer experimen-tation suggested a correction, the monotone conjecture [17, 19], that appears to be validfor flag manifolds of type A. Eremenko, Gabrielov, Shapiro, and Vainshtein [7] proved a re-sult that implies the monotone conjecture for some manifolds of two-step flags and concernscodimension-two subspaces that meet flags which are secant to the rational normal curve
Mathematics Subject Classification.
Key words and phrases.
Shapiro conjecture, Schubert calculus, flag manifold.Research supported in part by NSF grants DMS-0701050, DMS-0915211, and DMS-1001615.Research of Hillar supported in part by an NSF Postdoctoral Fellowship and an NSA Young Investigatorgrant.This research conducted in part on computers provided by NSF SCREMS grant DMS-0922866. along disjoint intervals. This suggested the secant conjecture, which asserts that an intersec-tion of Schubert varieties in a Grassmannian is transverse with all points real, provided thatthe Schubert varieties are defined by flags secant to a rational normal curve along disjointintervals. This was posed and evidence was presented for its validity in [10].The monotone secant conjecture is a common extension of both the monotone conjectureand the secant conjecture. It is also the last of the conjectures our group has made concerningreality in Schubert calculus of osculating flags. In addition to those mentioned, there is aversion of the Shapiro conjecture for the orthogonal Grassmannian which was proven byPurbhoo [16], and a version for the Lagrangian Grassmannian described in [21, Ch. 14.2].Exploratory computations in other flag manifolds suggest there is no regularity in the numberof real solutions to Schubert calculus problems given by osculating flags.We give here an open instance of the monotone secant conjecture, expressed as a systemof polynomial equations in local coordinates for the variety of flags E ⊂ E in C , wheredim E i = i . Let x , . . . ,x be indeterminates and consider the polynomials(1.1) f ( s,t,u ; x ) := det x x x x x x s s s s t t t t u u u u , g ( v,w ; x ) := det x x x x x x x x v v v v w w w w , which depend upon parameters s,t,u and v,w respectively. Conjecture 1.1.
Let s < t < u < · · · < s < t < u < v < w < · · · < v < w be realnumbers. Then the system of polynomial equations f ( s ,t ,u ; x ) = f ( s ,t ,u ; x ) = f ( s ,t ,u ; x ) = f ( s ,t ,u ; x ) = 0(1.2) g ( v ,w ; x ) = g ( v ,w ; x ) = g ( v ,w ; x ) = g ( v ,w ; x ) = 0 has twelve solutions, and all of them are real. These equations have geometric meaning. Let E be the span of the first two rows of eithermatrix and E the span of the first three rows of the second matrix that defines g . Then E ⊂ E is a general flag. If we let F ( s,t,u ) be the span of the last three rows of the matrixfor f , then this is a 3-plane that is secant to the rational normal curve γ : y (1 ,y,y ,y ,y )at the points γ ( s ) ,γ ( t ) ,γ ( u ). The equation f ( s,t,u ; x ) = 0 is the condition that E meet F ( s,t,u ) non-trivially. Smilarly, if F ( u,v ) is the span of the last two rows of the secondmatrix, which is 2-plane secant to γ at γ ( u ) and γ ( v ), then the equation g ( v,w ; x ) = 0 is thecondition that E meet F ( u,v ) non-trivially.The monotonicity hypothesis is that the four 3-planes given by s i ,t i , u i are secant alongintervals [ s i ,u i ] which are pairwise disjoint and occur before the pairwise disjoint intervals[ v i , w i ] where the 2-planes are secant. If the order of the intervals [ s ,u ] and [ v , w ] isreversed, the evaluation is no longer monotone. We tested 3 , ,
000 instances of Conjec-ture 1.1, finding only real solutions. In contrast, we tested 21 , ,
000 with the monotonicitycondition violated, of which 18 , ,
537 had some non real solutions.
ONOTONE SECANT CONJECTURE 3
We formulate the monotone secant conjecture, explain its relation to the other realityconjectures, describe data supporting it from a large computational experiment, and discusssome features observed in our data that go beyond the monotone secant conjecture. Thisexperiment verified the monotone secant conjecture in each of the 768,846,000 instances ittested. We have created a website [9] for viewing the data online. This includes pages forbrowsing the data and viewing the results for each Schubert problem. We only sketch theother reality conjectures, for they are described in the cited literature, and we also only sketchthe design and execution of this experiment, for the purpose of the paper [11] was to presentthe software environment we developed for such distributed computational experiments.This paper is organized as follows. In Section 2 we illustrate the main point of themonotone secant conjecture through the classical problem of four lines. Section 3 is a primeron flag manifolds and contains a precise statement of the monotone secant conjecture whilealso explaining its relation to the Shapiro, secant, and monotone conjectures. In Section 4 weexpand on the relation between the monotone secant and monotone conjectures, discuss theexperimental evidence for the monotone secant conjecture, and some phenomena we observedin our data. Lastly, in Section 5 we sketch the methods we used to test the conjecture.2.
The problem of four lines
The classical problem of four lines asks for the finitely many lines m that meet four givengeneral lines ℓ , ℓ , ℓ , ℓ in (projective) three-space. This has a pleasing synthetic solution,which leads to the first interesting case of the monotone secant conjecture.Three general lines ℓ , ℓ , ℓ lie in one ruling of a doubly-ruled quadric surface Q , withthe other ruling consisting of all lines that meet the first three. The line ℓ meets Q in twopoints, and through each of these points there is a line in the second ruling. These two lines, m and m , are the solutions to this problem. If the lines ℓ , ℓ , ℓ , ℓ are real, then so is Q , but the intersection of Q with ℓ is either two real points or a pair of complex conjugatepoints. In the first case, the problem of four lines has two real solutions, while in the second,it has no real solutions.Let us consider a variant in the manifold of flags consisting of a line m lying on a plane M in 3-space, m ⊂ M . Consider the Schubert problem in which m meets three lines ℓ , ℓ , ℓ and M contains two points, p and q . Then M contains the affine span p,q of p and q . Since m ⊂ M , we must have that m also meets p,q and is therefore a solution to the problem offour lines given by ℓ , ℓ , ℓ and p,q . As M is spanned by m and p,q , we see that solvingthis auxiliary problem of four lines solves our original Schubert problem. Furthermore, if thelines and points are real, then a solution m ⊂ M is real if and only if m is real.Suppose that the three lines are secant to a rational normal curve γ along disjoint intervalsand the points are p = γ ( s ) and q = γ ( t ), which do not lie in any interval of secancy. Thereare two possible combinatorial placements of the two points. Removing the three intervalsof secancy from γ results in three disjoint intervals along γ ≃ RP . Either both points γ ( s )and γ ( t ) lie in the same interval or they lie in different intervals. We examine each case. HEIN, HILLAR, MART´IN DEL CAMPO, SOTTILE, AND TEITLER
Fixing secant lines ℓ , ℓ , ℓ , the quadric Q described above is a hyperboloid of one sheet.This is displayed in Figures 1 and 2, along with γ and the lines. Suppose that γ ( s ) and γ ( t )lie in the same interval, say I , as indicated in Figure 1. Then the secant line they span, γ ❍❍❍❥ ✻ I Q
Figure 1.
The problem of four secant lines. ℓ ( s,t ), lies in the direction of I and meets the hyperboloid Q in two real points. Thus, inthis first case, our Schubert problem has two real solutions. (This is also an instance of thesecant conjecture, which holds for this problem of four lines [10, § γ ( s ) and γ ( t ) do not lie in the same interval, it ispossible to have no real solutions. Consider the choice of points γ ( s ) and γ ( t ) as shownin Figure 2, so that the secant line ℓ ( s,t ) does not meet the quadric Q . By our previous γ ❍❍❍❥ ✲ ℓ ( s,t ) Q Figure 2.
A non-monotone evaluation.analysis, there will be no real lines m meeting these four secant lines, and therefore no realsolutions m ⊂ M to our Schubert problem.We conclude that the positions of the points γ ( s ) ,γ ( t ) relative to the other intervals ofsecancy may affect whether or not the solutions are real. The schematic in Figure 3 illustratesthe relative positions of the secancies along γ (which is homeomorphic to the circle). Theidea behind the monotone secant conjecture is to attach to each interval the dimension ofthat part of the flag which it affects. This is 1 for m and 2 for M . The schematic on the lefthas labels 1 , , , ,
2, reading clockwise, starting just past the point s , while that on the rightreads 1 , , , ,
2. The labels increase monotonically in the first and do not in the second.
ONOTONE SECANT CONJECTURE 5 s t
All solutions real s t
Not all solutions real
Figure 3.
Schematic for the secancies.3.
Background
We develop the background for the statement of the monotone secant conjecture, definingflag varieties and their Schubert problems. More may be found in the book of Fulton [8].Fix positive integers a • := ( a < · · · < a k ) and n with a k < n . A flag E • of type a • is asequence of subspaces E • : { } ⊂ E a ⊂ E a ⊂ · · · ⊂ E a k ⊂ C n , where dim( E a i ) = a i . The set of all such flags forms the flag manifold F ℓ ( a • ; n ), which has dimension dim( a • ) := P ki =1 ( n − a i )( a i − a i − ), where a := 0. When a • = ( a ) is a singleton, F ℓ ( a • ; n ) is the Grassmannian of a -planes in C n , written Gr( a,n ). Flags of type 1 < < · · · < n − C n are complete .The positions of flags relative to a fixed complete flag F • stratify F ℓ ( a • ; n ) into topologicalcells whose closures are Schubert varieties . These positions are indexed by certain permu-tations. The descent set δ ( σ ) of a permutation σ ∈ S n is the set of numbers i such that σ ( i ) > σ ( i +1). For a permutation σ ∈ S n with δ ( σ ) ⊂ a • , the Schubert variety X σ F • is X σ F • = { E • ∈ F ℓ ( a • ; n ) | dim E a i ∩ F j ≥ { l ≤ i | j + σ ( l ) > n } ∀ i,j } . Flags E • in X σ F • have position σ relative to F • . A permutation σ with descent set containedin a • is a Schubert condition on flags of type a • . The Schubert variety X σ F • is irreduciblewith codimension ℓ ( σ ) := |{ i < j | σ ( i ) > σ ( j ) }| . A Schubert problem for F ℓ ( a • ; n ) is a list σ := ( σ , . . . , σ m ) of Schubert conditions for F ℓ ( a • ; n ) satisfying ℓ ( σ )+ · · · + ℓ ( σ m ) = dim( a • ).Given a Schubert problem σ for F ℓ ( a • ; n ) and complete flags F • , . . . , F m • , the intersection(3.1) X σ F • ∩ · · · ∩ X σ m F m • is an instance of σ . When the flags are in general position, this intersection is transverse andzero-dimensional [13], and it consists of all flags E • ∈ F ℓ ( a • ; n ) having position σ i relative to F i • , for each i = 1 , . . . , m . Such a flag E • is a solution to this instance of σ .The degree of a zero-dimensional intersection (3.1) is independent of the choice of the flagsand we call this number d ( σ ) the degree of the Schubert problem σ . When the intersectionis transverse, the number of solutions to σ equals its degree. HEIN, HILLAR, MART´IN DEL CAMPO, SOTTILE, AND TEITLER
When the flags F • , . . . , F m • are real, the solutions to the Schubert problem need not bereal. The monotone secant conjecture posits a method to select the flags F • so that allsolutions are real, for a certain class of Schubert problems.Let γ : R → R n be a rational normal curve, which is any curve affinely equivalent to themoment curve t (1 ,t,t , . . . ,t n − ). Consider this projectively, so that γ is homeomorphicto RP , which is a circle. A flag F • is secant along an interval I of γ if every subspace in theflag is spanned by its intersection with I . A list of flags F • , . . . ,F m • secant to γ is disjoint if the intervals of secancy are pairwise disjoint. Disjoint flags are naturally ordered by theorder in which their intervals of secancy lie within RP . We remark that this order is to betaken cyclically as in Figure 3, and with respect to one of the two orientations of RP .A permutation σ is Grassmannian if it has a unique descent, for these, let δ ( σ ) be thedescent. A Grassmannian Schubert problem is one that involves only Grassmannian Schu-bert conditions. A list of disjoint secant flags F • , . . . ,F m • is monotone with respect to aGrassmannian Schubert problem ( σ , . . . ,σ m ) if the function F i • δ ( σ i ) is monotone withrespect to one of the two orientations of RP . In other words, if δ ( σ i ) < δ ( σ j ) = ⇒ F i < F j , for all i,j , where < is induced by one of the two cyclic orderings of RP . Monotone Secant Conjecture 3.1.
For any Grassmannian Schubert problem ( σ , . . . , σ m ) on the flag manifold F ℓ ( a • ; n ) and any disjoint secant flags F • , . . . ,F m • that are monotonewith respect to the Schubert problem, the intersection X σ F • ∩ X σ F • ∩ · · · ∩ X σ m F m • is transverse with all points real. Conjecture 1.1 is the monotone secant conjecture for a Schubert problem on F ℓ (2 ,
3; 5)involving the Schubert conditions σ := 13245 and τ := 12435, where we write permutationsin one-line notation, so that σ (2) = 3 and τ (2) = 2. Then δ ( σ ) = 2, δ ( τ ) = 3, and ℓ ( σ ) = ℓ ( τ ) = 1, so that ( σ,σ,σ,σ,τ,τ,τ,τ ) is a Schubert problem for F ℓ (2 ,
3; 5), as dim( F ℓ (2 ,
3; 5)) = 8.We use exponential notation for repeated conditions, so that this Schubert problem is writtenas ( σ ,τ ). The corresponding Schubert varieties are X σ F • = { E • ∈ F ℓ (2 ,
3; 5) | dim E ∩ F ≥ } ,X τ F • = { E • ∈ F ℓ (2 ,
3; 5) | dim E ∩ F ≥ } , that is, the set of flags E • whose 2-plane E meets a fixed 3-plane F non-trivially, and theset of E • where E meets a fixed 2-plane F non-trivially, respectively. Consequently, wewrite X σ F for X σ F • and X τ F for X τ F • .For s,t,u,v,w ∈ R , let F ( s,t,u ) be the linear span of γ ( s ) , γ ( t ), and γ ( u ) and F ( v,w )be the linear span of γ ( v ) and γ ( w ); these are a secant 3-plane and a secant 2-plane tothe rational normal curve, respectively. The condition f ( s,t,u ; x ) = 0 of Conjecture 1.1 isequivalent to the membership E • ∈ X σ F ( s,t,u ). Similarly, the condition g ( v,w ; x ) = 0 isequivalent to the membership E • ∈ X τ F ( v,w ). Lastly, the condition on the ordering of ONOTONE SECANT CONJECTURE 7 the points s i ,t i ,u i ,v j ,w j in Conjecture 1.1 implies that the relevant subspaces F ( s i ,t i ,u i ) and F ( v j ,w j ) for i,j = 1 , . . . , a ; n ), anylist of disjoint secant flags F • , . . . ,F m • is monotone with respect to any Schubert problem( σ , . . . ,σ m ), as all the conditions have the same descent. In this way, the monotone secantconjecture reduces to the secant conjecture. Secant Conjecture 3.2.
For any Schubert problem ( σ , . . . , σ m ) on any Grassmannian andany disjoint secant flags F • , . . . , F m • , the intersection X σ F • ∩ X σ F • ∩ · · · ∩ X σ m F m • is transverse with all points real. We studied this conjecture in a large-scale experiment whose results are described in [10],solving 1,855,810,000 instances of 703 Schubert problems on 13 different Grassmannians,verifying the secant conjecture in each of the 448,381,157 instances checked. This took 1.065teraHertz-years of computing.The osculating flag F • ( t ) is the flag whose j -dimensional subspace F j ( t ) is the span ofthe first j derivatives γ ( t ) ,γ ′ ( t ) , . . . ,γ ( j − ( t ) of γ at t . This subspace F j ( t ) is the unique j -dimensional subspace having maximal order of contact, namely j , with γ at γ ( t ). It followsthat the limit of any family of flags whose intervals of secancy shrink to a point γ ( t ) is thisosculating flag F • ( t ). In this way, the limit of the monotone secant conjecture, as the secantflags become osculating flags, is a similar conjecture where we replace monotone secant flagsby monotone osculating flags. Monotone Conjecture 3.3.
For any Schubert problem ( σ , . . . , σ m ) on the flag manifold F ℓ ( a • ; n ) and any flags F • , . . . , F m • osculating a rational normal curve γ at real points thatare monotone with respect to the Schubert problem, the intersection X σ F • ∩ X σ F • ∩ · · · ∩ X σ m F m • is transverse with all points real. Ruffo, et al. [17] formulated and studied this conjecture, establishing special cases andgiving substantial experimental evidence in support of it.The Shapiro conjecture is a specialization of the monotone secant conjecture that bothrestricts to the Grassmannian and to osculating flags. This was posed around 1995 byBoris Shapiro and Michael Shapiro and studied in [18]. Proofs were given by Eremenko andGabrielov for Gr( n − n ) [5, 6] using complex analysis and in complete generality by Mukhin,Tarasov, and Varchenko [14, 15] using integrable systems and representation theory. Shapiro Conjecture 3.4.
For any Schubert problem ( σ , . . . , σ m ) in a Grassmannian andany distinct real numbers t , . . . , t m , the intersection X σ F • ( t ) ∩ X σ F • ( t ) ∩ · · · ∩ X σ m F • ( t m ) HEIN, HILLAR, MART´IN DEL CAMPO, SOTTILE, AND TEITLER is transverse with all points real. Results
A consequence of the example discussed in Section 2 is that the secant conjecture (like theShapiro conjecture before it) does not hold for flag manifolds F ℓ ( a • ; n ) that are not Grass-mannians. The monotonicity condition appears to correct this failure in both conjectures.We give more details on the relation of the monotone conjecture to the monotone secantconjecture and give a conjecture that interpolates between the two. Then we discuss someof our data in an experiment that tested both conjectures.4.1. The monotone conjecture is the limit of the monotone secant conjecture.
The osculating plane F i ( s ) is the unique i -dimensional subspace having maximal order ofcontact with the rational normal curve γ at the point γ ( s ), and therefore it is a limit ofsecant planes. We give a more precise statement of this fact. Proposition 4.1.
Let { s ( j )1 . . . , . . . , s ( j ) i } for j = 1 , , . . . be a sequence of lists of i distinctcomplex numbers with the property that for each p = 1 , . . . , i , we have lim j →∞ s ( j ) p = s , for some number s . Then, lim j →∞ span { γ ( s ( j )1 ) , . . . , γ ( s ( j ) i ) } = F i ( s ) . As explained in the previous section, by this proposition, the monotone secant conjectureimplies the monotone conjecture. This has a partial converse which follows from a standardlimiting argument.
Theorem 4.2.
Let ( σ , . . . , σ m ) be a Schubert problem on F ℓ ( a ; n ) for which the monotoneconjecture holds. Then, for any distinct real numbers that are monotone with respect to ( σ , . . . ,σ m ) , there exists a number ǫ > such that, if for each i = 1 , . . . ,m , F i • is a flagsecant to γ along an interval of length ǫ containing t i , then the intersection X σ F • ∩ X σ F • ∩ · · · ∩ X σ m F m • is transverse with all points real. Generalized monotone secant conjecture.
We generalize the monotone secant con-jecture, replacing secant flags by flags which are spanned by osculating subspaces, as in [10, § generalized secant subspace to the rational normal curve γ is a subspace that is spannedby subspaces osculating γ at real points. This notion includes secant subspaces as well asosculating subspaces, as a point of γ generates a one-dimensional subspace osculating γ . A generalized secant flag is one consisting of generalized secant subspaces. A generalized secant ONOTONE SECANT CONJECTURE 9 flag is secant along an interval I if the osculating subspaces spanning its subspaces osculate γ at points of I . Conjecture 4.3 (Generalized monotone secant conjecture) . For any Grassmannian Schubertproblem ( σ , . . . , σ m ) on the flag manifold F ℓ ( a • ; n ) and any disjoint generalized secant flags F • , . . . ,F m • that are monotone with respect to the Schubert problem, the intersection X σ F • ∩ X σ F • ∩ · · · ∩ X σ m F m • is transverse with all points real. This conjecture contains the monotone and monotone secant conjectures as special cases,and interpolates between the two.4.3.
Experimental evidence for the monotone secant conjecture.
While its rela-tion to existing conjectures led to the monotone secant conjecture, we believe the immenseweight of empirical evidence is the strongest support for it. We conducted an experimentthat tested 11,141,897,000 instances of 1300 Schubert problems on 19 flag manifolds. Ofthese, 768,846,000 were instances of the monotone secant conjecture, which was verifiedin every case tested. We also tested 918,902,000 instances of the monotone conjecture forcomparison. The remaining instances involved non-monotone evaluations of either disjointsecant flags or osculating flags. Our data consistently displayed a striking inner border, anda number of Schubert problems exhibited lower bounds on their numbers of real solutions.This experiment used 1.9 teraHertz-years of computing.Table 1 shows the data we obtained for the Schubert problem ( σ ,τ ) with 12 solutions onthe flag manifold F ℓ (2 ,
3; 5) introduced in Conjecture 1.1. We computed 24,000,000 instances
Real Solutions N ec k l a ce Table 1.
Necklaces vs. real solutions for ( σ ,τ ) in F ℓ (2 ,
3; 5).of this problem, all involving flags that were secant to the rational normal curve along disjointintervals. This took 17.618 gigaHertz-years. The columns are indexed by even integers from0 to 12, indicating the number of real solutions. The rows are indexed by necklaces , which aresequences δ ( σ ) , . . . ,δ ( σ m ), where δ ( σ i ) denotes the descent of the Grassmannian permutation σ i , as described in Section 3. In the table a 2 represents the condition on the two-plane E given by the permutation σ = 13245, and a 3 represents the condition on E given by thepermutation τ = 12435.In Table 1, the first row labeled with 22223333 represents tests of the monotone secantconjecture, since the only entries are in the column for 12 real solutions, the monotone secantconjecture was verified in 3,000,000 instances. This is the only row with only real solutions.Compare this to the 16,000,000 instances of the same Schubert problem, but with oscu-lating flags, which we present in Table 2. This computation took 85.203 gigahertz-days.Both tables are similar with nearly identical “inner borders”, except for the shaded box Real Solutions N ec k l a ce Table 2.
Necklaces vs. real solutions for ( σ ,τ ) in F ℓ (2 ,
3; 5).in Table 2. In fact, by a standard argument similar to that which implied Theorem 4.2,we may conclude that every number of real solutions to a Schubert problem observed for agiven necklace with osculating flags also occurs for that Schubert problem and necklace withsecant flags, where the points of secancy are sufficiently clustered. That is, the support of atable for the monotone conjecture will be a subset of the support of the corresponding tablefor the monotone secant conjecture. There are some Schubert problems for which we didnot observe this containment; the reason for this is that we aparently did not compute aninstance with secant flags whose points of secancy were sufficiently clustered.4.4.
Lower bounds and inner borders.
The most enigmatic phenomenon that we observein our data is the presence of an “inner border” for many geometric problems, as we havepointed out in example of Table 1. That is, for some necklaces (besides the monotone ones),there appears to be a lower bound on the number of real solutions. We do not understandthis phenomenon, even conjecturally. Our software that displays the tables is designed tohighlight this feature of our data.Another common phenomenon is that for many problems, there are always at least somereal solutions, for any necklace. (Note that the last rows of Tables 1 and 2 had instances withno real solutions). Table 3 displays an example of this for a Schubert problem ( σ ,τ ) on F ℓ (2 ,
3; 6) with 21 solutions, where σ := 142356 has δ ( σ ) = 2 and ℓ ( σ ) = 2 and τ := 124356has δ ( τ ) = 3 and ℓ ( τ ) = 1. Very prominently, it appears that at least 11 of the solutionswill always be real. This computation took 7.67 gigaHertz-years. ONOTONE SECANT CONJECTURE 11
Real Solutions N ec k l a ce Table 3.
Enumerative Problem W X = 21 on F ℓ (2 ,
3; 6)Such lower bounds and inner borders were observed when studying the monotone con-jecture [17]. Eremenko and Gabrielov established lower bounds for the Wronski map [4] inSchubert calculus for the Grassmannian, more recently Azar and Gabrielov [1] establishedlower bounds for some instances of the monotone conjecture which were observed in [17].5.
Methods
Our experimentation was possible as instances of Schubert problems are simple to modelon a computer. The procedure we use may be semi-automated and run on supercomputers.We will not describe how this automation is done, for that is the subject of the paper [11];instead, we explain here the computations we performed.For a Schubert condition σ on a flag variety F ℓ ( a • ; n ), let j ( σ ) be the dimension of thelargest subspace in a flag F • that is needed to define X σ F • . For example, we have seen that j (13245) = 3 and j (12435) = 2.To compute an instance of a Schubert problem ( σ , . . . , σ m ) corresponding to a necklace ν , we first select N := N ( σ ) + . . . + N ( σ m ) real numbers and then group them into disjointsubsets s (1) , . . . ,s ( m ) where s ( i ) consists of N ( σ i ) consecutive numbers. Furthermore, therelative ordering of the subsets corresponds to the necklace ν . Having done this, each subset s ( i ) defines a secant (partial) flag F • ( s ( i ) ). We use these flags to formulate the instance ofthe Schubert problem X σ F • ( s (1) ) ∩ X σ F • ( s (2) ) ∩ · · · ∩ X σ m F • ( s ( m ) )as a system of polynomials in dim( a • ) local coordinates for F ℓ ( a • ; n ), whose common zeroesrepresent the solutions to this instance of the Schubert problem. This was illustrated in theIntroduction when Conjecture 1.1 was presented. See [8, 17, 18] for details.Given this system of polynomials, we use Gr¨obner bases to compute a polynomial in onevariable of minimal degree in the ideal of these equations. This univariate polynomial is calledan eliminant . If the eliminant is square-free and has degree equal to the expected numberof complex solutions (this is easily verified) then the number of real roots of the eliminantequals the number of real solutions to the Schubert problem. This follows from the form of alexicographic Gr¨obner basis for the ideal, as described by the Shape Lemma [3]. This is givenin more detail in § problem by computing the number of real roots of the eliminant. For this, we use MAPLE’s realroot command, which uses symbolic methods based on Sturm sequences to determinethe number of real roots of a univariate polynomial. If the software is reliably implemented,which we believe, then this computation provides a proof that the given instance has thecomputed number of real solutions to the original Schubert problem.In our computations, for a given Schubert problem, we first make a choice of N realnumbers, and then use these same N numbers for all necklaces for that problem. Then wemake another choice, and so on, ultimately making thousands to millions of such choices.For each Schubert problem we studied, we not only tested many instances of the monotonesecant conjecture, but also of the monotone conjecture, comparing the two as we did for theSchubert problem ( σ ,τ ) in F ℓ (2 ,
3; 5), where σ = 13245 and τ = 12435. To computeinstances of the monotone conjecture, we choose real numbers s , . . . ,s m and use osculatingflags F • ( s ) , . . . ,F • ( s m ). This is also described in [17, § F • ( ∞ ) osculating the rational normal curveat infinity. Then we used local coordinates for X σ F • ( ∞ ), in place of the local coordinatesfor F ℓ ( a • ; n ); this uses ℓ ( σ i ) fewer local coordinates.For some Schubert problems we wanted to study, there were several hundred to manythousands of necklaces, and for these we uniformly chose a much smaller set of necklaces tocompute, which we called coarse necklaces. In our on-line tables, we encoded these choicesin a variable called computation type. Computation types 4 and 7 were for instances ofthe monotone conjecture, 5 and 8 for the monotone secant conjecture, and 6 and 9 for thegeneralized monotone secant conjecture. In each of these, the first number indicates that weused all necklaces, while the second we used coarse necklaces. References [1] Monique Azar and Andrei Gabrielov, Some lower bounds in the B. and M. Shapiro conjecture for flagvarieties, Discrete Comput. Geom. (2011), no. 4, 636–659.[2] D.J. Bates, F. Bihan, and F. Sottile, Bounds on the number of real solutions to polynomial equations,Int. Math. Res. Not. IMRN (2007), no. 23, Art. ID rnm114, 7.[3] E. Becker, M.G. Marinari, T. Mora, and C. Traverso, The shape of the Shape Lemma, Proc. ISSAC’94, ACM Press, New York, 1994.[4] A. Eremenko and A. Gabrielov, Degrees of real Wronski maps, Discrete Comput. Geom. (2002),no. 3, 331–347.[5] A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiroconjecture in real enumerative geometry, Ann. of Math. (2) (2002), no. 1, 105–129.[6] A. Eremenko and A. Gabrielov, An elementary proof of the B. and M. Shapiro conjecture for rationalfunctions, Notions of positivity and the geometry of polynomials, Springer, Basel, 2011, pp. 167–178.[7] A. Eremenko, A. Gabrielov, M. Shapiro, and A. Vainshtein, Rational functions and real Schubertcalculus, Proc. Amer. Math. Soc. (2006), no. 4, 949–957 (electronic).[8] W. Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge UniversityPress, Cambridge, 1997. ONOTONE SECANT CONJECTURE 13 [9] L. Garc´ıa-Puente, J. Hauenstein, N. Hein, C. Hillar, A. Mart´ın del Campo, J. Ruffo, F. Sottile, andZ. Teitler, Frontiers of reality in Schubert calculus, 2010, .[10] L.D. Garc´ıa-Puente, N. Hein, C. Hillar, A. Mart´ın del Campo, J. Ruffo, F. Sottile, and Z. Teitler, TheSecant Conjecture in the real Schubert calculus, Experimental Math., 21 (2012), pp. 252–265.[11] C. Hillar, L. Garc´ıa-Puente, A. Mart´ın del Campo, J. Ruffo, Z. Teitler, S. L. Johnson, and F. Sottile,Experimentation at the frontiers of reality in Schubert calculus, Gems in Experimental Mathematics,Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010, pp. 365–380.[12] A.G. Khovanskii, Fewnomials, Trans. of Math. Monographs, 88, AMS, 1991.[13] S. L. Kleiman, The transversality of a general translate, Compositio Math. (1974), 287–297.[14] E. Mukhin, V. Tarasov, and A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometryand the Bethe ansatz, Ann. of Math. (2) (2009), no. 2, 863–881.[15] E. Mukhin, V. Tarasov, and A. Varchenko, Schubert calculus and representations of the general lineargroup, J. Amer. Math. Soc. (2009), no. 4, 909–940.[16] K. Purbhoo, Reality and transversality for Schubert calculus in OG( n, n + 1), Math. Res. Lett. (2010), no. 6, 1041–1046.[17] J. Ruffo, Y. Sivan, E. Soprunova, and F. Sottile, Experimentation and conjectures in the real Schubertcalculus for flag manifolds, Experiment. Math. (2006), no. 2, 199–221.[18] F. Sottile, Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Ex-periment. Math. (2000), no. 2, 161–182.[19] F. Sottile, Some real and unreal enumerative geometry for flag manifolds, Michigan Math. J. (2000),573–592.[20] F. Sottile, Frontiers of reality in Schubert calculus, Bull. Amer. Math. Soc. (N.S.) (2010), no. 1,31–71.[21] F. Sottile, Real solutions to equations from geometry, University Lecture Series, vol. 57, AmericanMathematical Society, 2011. Nickolas Hein, Department of Mathematics, University of Nebraska at Kearney, Kear-ney, Nebraska 68849, USA
E-mail address : [email protected] URL : Christopher J. Hillar, Mathematical Sciences Research Institute, 17 Gauss Way, Berke-ley, CA 94720-5070, USA
E-mail address : [email protected] URL : Abraham Mart´ın del Campo, IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
E-mail address : [email protected] URL : http://pub.ist.ac.at/~adelcampo/ Frank Sottile, Department of Mathematics, Texas A&M University, College Station,Texas 77843, USA
E-mail address : [email protected] URL : Zach Teitler, Department of Mathematics, Boise State University, Boise, Idaho 83725,USA
E-mail address : [email protected] URL ::