REAL SCHUBERT CALCULUS: POLYNOMIAL SYSTEMS AND A CONJECTURE OF SHAPIRO AND SHAPIRO

FRANK SOTTILE

Abstract. Boris Shapiro and Michael Shapiro have a conjecture concerning the Schubert calculus and real enumerative geometry and which would give infinitely many families of zero-dimensional systems of real polynomials (including families of overdetermined systems)—all of whose solutions are real. It has connections to the pole placement problem in linear systems theory and to totally positive matrices. We give compelling computational evidence for its validity, prove it for infinitely many families of enumerative problems, show how a simple version implies more general versions, and present a counterexample to a general version of their conjecture.

This is a companion paper to [15] and [47], which describe the mathematics involved in two spectacular computations verifying specific instances of this conjecture.

1. Introduction

Determining the number of real solutions to a system of polynomial equations is a challenging problem in symbolic and numeric computation [18, 44] with real world applications. Related questions include when a problem of enumerative geometry can have all solutions real [39] and when may a given physical system be controlled by real output feedback [6, 32, 45]. In May 1995, Boris Shapiro and Michael Shapiro communicated to the author a remarkable conjecture connecting these three lines of inquiry.

They conjectured a relation between topological invariants of the real and of the complex points in an intersection of Schubert cells in a flag manifold, if the cells are chosen according to a recipe they give. When the intersection is zero-dimensional, this asserts that all points are real. Their conjecture is false—we give full description and present a counterexample in Section 5. However, there is considerable evidence for their conjecture if the Schubert cells are in a Grassmann manifold. It is this variant which is related to the lines of inquiry above and which this paper is about.

Here is the simplest (but still very interesting and open) special case of this conjecture: Let m, p > 1 be integers and let X be a p × m-matrix of indeterminates. Let K(s) be the m × (m + p)-matrix of polynomials in s whose i, jth entry is

Date: 1 December 1998.

1991 Mathematics Subject Classification. 12D10, 13P10, 14N10, 14M15, 14Q20, 14P99, 15A48, 93B55.

Key words and phrases. Enumerative geometry, Grassmannian, Gröbner basis, Overdetermined system.

Supported in part by NSERC grant OGP0170279 and NSF grants DMS-9022140 and DMS-9701755.

\binom{j − i}{i − 1}s^{j−i}. (1)

Set

φ_{m,p}(s; X) := det [ \begin{matrix}K(s) \\ I_{p} X \end{matrix} ] ,

where I_{p} is the p × p identity matrix.

Conjecture 1.1 (Shapiro-Shapiro). For all integers m, p > 1, the polynomial system

φ_{m,p}(1; X) = φ_{m,p}(2; X) = ⋯ = φ_{m,p}(mp; X) = 0 (2)

is zero-dimensional with

d_{m,p} := \frac{1! 2! 3! ⋯ (p−2)! (p−1)! · (mp)!}{m! (m+1)! (m+ 2)! ⋯ (m+p−1)!} (3)

solutions, and all of them are real.

It is a Theorem of Schubert [36] that d_{m,p} is a sharp bound for the number of isolated solutions. Conjecture 1.1 has been verified for all 1 < m ≤ p with mp ≤ 12. The case of (m, p) = (3, 4) (d_{m,p} = 462) is due to an heroic calculation of Faugère, Rouillier, and Zimmermann [15] (see Section 2.4 for a discussion).

Conjecture 1.1 is related to a question of Fulton [16, §7.2]: “How many solutions to a problem in enumerative geometry may be real, where that problem consists of counting figures of some kind having a given position with respect to some given (fixed) figures.”. For 2-planes having a given position with respect to fixed linear subspaces, the answer is that all may be real [40]. This was also shown for the problem of 3264 plane conics tangent to five given conics [31]. More examples, including that of 3-planes in ℂ^{6} meeting 9 given 3-planes nontrivially, are found in [39, 41]. The result [15] extends this to 3-planes in ℂ^{7} meeting 12 given 4-planes nontrivially.

Only the simplest form of the conjecture of Shapiro and Shapiro has appeared in print [23, 33, 39]. While more general forms have circulated informally, there is no definitive source describing the conjectures or the compelling evidence that has accumulated (or a counterexample to the original conjecture). The primary aim of this paper is to rectify this situation and make these conjectures available to a wider audience.

In Section 2, we describe a version of the conjecture related to the pole placement problem of linear systems theory. For this, the integers 1, 2, …, mp in the polynomial system (2) of Conjecture 1.1 are replaced by generic real numbers and all d_{m,p} solutions are asserted to be real. We present evidence (computational and Theorems) in support of it. Subsequent sections describe the conjecture in greater generality— for enumerative problems arising from the Schubert calculus on Grassmannians in Section 3 and a newer extension involving totally positive matrices [1] in Section 4. We describe and give evidence for each extension and show how the version of the conjecture in Section 2 implies more general versions involving Pieri-type enumerative problems. In Section 5, we present a counterexample to their original conjecture and discuss further questions.

A remark on the form of these conjectures is warranted. Conjecture 1.1 gives an infinite list of specific polynomial systems, and conjectures that each has only real solutions. The full conjectures are richer. For each collection of Schubert data, Shapiro and Shapiro give a continuous family of polynomial systems and conjecture that each of the resulting systems of polynomials has only real solutions. Conjecture 1.1 concerns one specific polynomial system in each family, for an infinite subset of Schubert data.

Results here were aided or are due to computations. Further documentation including Maple V.5 and Singular 1.2.1 [19] scripts used are available on the web page [42].

Contents

1. Introduction 1 2. Linear equations in Plücker coordinates 3 2.1. Some enumerative geometry 3 2.2. The conjecture of Shapiro and Shapiro 3 2.3. Pole placement problem 5 2.4. Computational evidence 6 2.5. Equivalent systems 7 2.6. Conjecture ?? for m = 2, p = 3 9 3. Schubert conditions on a Grassmannian 11 3.1. The Schubert calculus on Grass(p, m + p) 11 3.2. Systems of polynomials 12 3.3. Pieri Schubert conditions 13 3.4. An infinite family 17 3.5. Computational evidence 19 4. Total positivity 21 5. Further remarks 24 5.1. A counterexample to the original conjecture 24 5.2. Further questions 25 Acknowledgements 27 References 28

2. Linear equations in Plücker coordinates

2.1. Some enumerative geometry. Consider the following problem in enumerative geometry: How many p-planes meet mp general m-planes in ℂ^{m+p} nontrivially?

The set of p-planes in ℂ^{m+p}, Grass(p, m + p), is called the Grassmannian of pplanes in ℂ^{m+p}. This complex manifold of dimension mp is an algebraic subvariety of the projective space ℙ^{\binom{m+p}{p}−1}. To see this, represent a p-plane X in ℂ^{m+p} as the row space of a p × (m + p)-matrix, also written X. The maximal minors of X are its Plücker coordinates and determine a point in ℙ^{\binom{m+p}{p}−1}. This gives the Plücker embedding of Grass(p, m + p). If X is generic, then its first p columns are linearly independent, so we may assume they form a p×p-identity matrix. The remaining mp entries determine X uniquely and give local coordinates for Grass(p, m + p), showing it has dimension mp.

Consider a m-plane K to be the row space of a m × (m + p)-matrix, also written K. Then K ∩ X is nontrivial if and only if

det [ \begin{matrix}K \\ X \end{matrix} ] = 0.

Laplace expansion along X gives a linear equation in the Plücker coordinates of X.

If K_{1}, …, K_{mp} are m-planes in general position, then the conditions that X meet each of the K_{i} nontrivially are mp linear equations in the Plücker coordinates of X, and these are independent by Kleiman’s Transversality Theorem [24]. Hence there are finitely many p-planes X which meet each K_{i} nontrivially and this number is the degree of Grass(p, m + p) in ℙ^{\binom{m+p}{p}−1}, which Schubert [36] determined to be d_{m,p}.

2.2. The conjecture of Shapiro and Shapiro. Shapiro and Shapiro gave a recipe for selecting real m-planes K_{1}, …, K_{mp} and conjecture that when they are in in general position, all d_{m,p} p-planes meeting each K_{i} are real. The standard rational normal curve is the image of the map γ : ℝ → ℝ^{m+p} given by

γ : s ⟼ (1, s, s^{2}, …, s^{m+p−1}) (4)

Then the matrix K(s) of the Introduction (1) has rows

γ(s), γ′(s), \frac{γ′′(s)}{2}, …, \frac{γ^{(m−1)}(s)}{(m − 1)!},

where we take derivatives with respect to the parameter s. Let X be a p × m-matrix of indeterminates. Define

φ_{m,p}(s; X) := det [ \begin{matrix}K(s) \\ I_{p} X \end{matrix} ] .

Conjecture 2.1 (Shapiro-Shapiro). For all integers m, p > 1 and almost all distinct real numbers s_{1}, …, s_{mp}, the system of mp equations

φ_{m,p}(s_{1}; X) = φ_{m,p}(s_{2}; X) = ⋯ = φ_{m,p}(s_{mp}; X) = 0 (5)

is zero-dimensional with d_{m,p} real solutions.

Let K(s) denote both the m × (m + p)-matrix defined above and its row space, an m-plane. Conjecture 2.1 asserts that the m-planes K(s_{1}), …, K(s_{mp}) are in general position, and any p-plane meeting each K(s_{i}) is real. The systems are zerodimensional [3, 11] and there are generically no multiplicities. Conjecture 1.1 is the special case when s_{i} = i.

Example 2.2. We establish Conjecture 2.1 when m = p = 2. Then

φ_{2,2}(s; X) = det [ \begin{matrix}1 & s & s^{2} & s^{3} \\ 0 & 1 & 2s & 3s^{2} \\ 1 & 0 & x_{11} & x_{12} \\ 0 & 1 & x_{21} & x_{22} \end{matrix} ]

is

s^{4} − 2s^{3}x_{21} + s^{2}x_{22} − 3s^{2}x_{11} + 2sx_{12} + x_{11}x_{22} − x_{12}x_{21}.

We show that if s, t, u, v ∈ ℝ are distinct, then the system of polynomial equations

φ_{2,2}(s) = φ_{2,2}(t) = φ_{2,2}(u) = φ_{2,2}(v) = 0 (6)

has all d_{2,2} = 2 solutions real. Our method will be to solve (6) by elimination.

Let e_{i} be the ith elementary symmetric polynomial in s, t, u, v. In the lexicographic term order with x_{11} > x_{12} > x_{22} > x_{21} on the ring ℚ(s, t, u, v)[x_{11}, x_{12}, x_{22}, x_{21}], the ideal ⟨φ_{2,2}(s), φ_{2,2}(t), φ_{2,2}(u), φ_{2,2}(v)⟩ has a Gröbner basis consisting of the following polynomials:

2x_{21} − e_{1}, x_{22} − 3x_{11} − e_{2}, 2x_{12} + e_{3}, and 12x_{11}^{2} + 4e_{2}x_{11} + e_{1}e_{3} − 4e_{4}.

Thus, for distinct s, t, u, v, the system (6) has 2 solutions and they are real if the discriminant of the last equation,

16e_{2}^{2} − 48e_{1}e_{3} + 192e_{4},

is positive. Expanding this discriminant in the parameters s, t, u, v, we obtain

8 ((s − t)^{2}(u − v)^{2} + (s − u)^{2}(t − v)^{2} + (s − v)^{2}(t − u)^{2}) .

Hence all solutions are real, establishing Conjecture 2.1 when m = p = 2. Theorem 2.3 proves Conjecture 2.1 when (m, p) = (2, 3).

2.3. Pole placement problem. Suppose we have a physical system (for example, a mechanical linkage) with inputs u ∈ ℝ^{m} and outputs y ∈ ℝ^{p} for which there are internal states x ∈ ℝ^{n} such that the system evolves by the first order linear differential equation

\begin{aligned}ẋ & = & Ax + Bu, \\ y & = & Cx. \end{aligned} (7)

(We assume n is the minimal number of internal states needed to obtain a first order equation.) If the input is controlled by constant output feedback, u = Xy, then we obtain

ẋ = (A + BXC)x.

The natural frequencies of this controlled system are the roots s_{1}, …, s_{n} of

φ(s) := det(sI_{n} − A − BXC). (8)

The pole assignment problem asks the inverse question: Given a system (7) and a polynomial φ(s) of degree n, which feedback laws X satisfy (8)?

A coprime factorization of the transfer function is two matrices N(s), D(s) of polynomials with det(D(s)) = det(sI_{n} − A) and N(s)D(s)^{−1} = C(sI_{n} − A)^{−1}B. This always exists. A standard transformation (cf. [6, §2]) shows that, up to a sign of ±1,

φ(s) = det [ \begin{matrix}N(s) & D(s) \\ I_{p} & X \end{matrix} ] , (9)

If we set K(s) := [N(s) D(s)], write K(s) for the m-dimensional row space of this matrix, and let X be the p-plane [I_{p} X], then (9) is equivalent to

X ∩ K(s_{i}) ≠ {0} for i = 1, …, n, (10)

where s_{1}, …, s_{n} are the roots of φ(s).

If the m-planes K(s_{1}), …, K(s_{n}) are in general position, then mp ≥ n is necessary for there to be any feedback laws. These m-planes are not a priori in general position.

To see this, let K : ℙ^{1} → Grass(m, m + p) be the extension of the map given by s ↦ K(s). Then K is a parameterized rational curve of degree n in Grass(m, m + p). The space of all such curves K with n distinguished points {K(s_{1}), …, K(s_{n})} has dimension [43]

mp + n(m + p) + n.

The space of all n-tuples of m-planes has dimension nmp. Therefore when

n > mp/(mp − m − p − 1),

such n-tuples constitute a proper subvariety of all n-tuples of m-planes.

However, the General Position Lemma [6] (see also [11]) states that there is a Zariski open subset of the data A, B, C, φ such that the m-planes K(s_{1}), …, K(s_{n}) are in general position in that the set of X satisfying (10) has dimension mp − n.

Since all rational curves K : ℙ^{1} → Grass(p, m + p) of degree n with K(∞) = [0 I_{p}] arise in this way [28], the polynomial systems of Conjecture 2.1 are instances of the pole placement problem. Interestingly, these very systems figure prominently in a proof of the General Position Lemma [4].

An important question is whether a given real system may be controlled by real feedback [5, 32, 33, 45, 48]: If all roots of φ(s) are real, are there any real feedback laws X satisfying (9)? Few specific examples have been computed [7, 29, 33, 48]. In [33] an attempt was made to gauge how likely it is for a real system to be controllable by real feedback and how many of the feedback laws are real—in the case of (m, p) = (2, 4) so that d_{m,p} = 14. In all, 600 different curves K(s) were generated, and each of these were combined with 25 polynomials φ(s) having 8 real roots. Only 7 of the resulting 15,000 systems had all feedback laws real. This is in striking contrast to the systems given in Conjecture 2.1, where all the feedback laws are conjectured to be real.

2.4. Computational evidence. Consider (9) as a map Grass(p, m + p) → ℙ^{mp} in local coordinates which associates a p-plane X to a polynomial φ (modulo scalars) of degree at most mp. When K(s) is the curve K_{m,p}(s) of Conjecture 2.1, the inverse image of the polynomial 1 is the single real point [0 I_{p}]. Rosenthal suggested that the fibre over a nearby polynomial may consist of d_{m,p} real points.

Inspired by this, Rosenthal and Sottile [33] tested and verified several thousand instances of Conjecture 2.1 when (m, p) = (2, 4). Each was a specific choice of m, p and mp distinct real numbers s_{1}, …, s_{mp} for which we showed all solutions to (5) are real. Any verified instance implies that all nearby instances in the space of parameters s_{1}, …, s_{mp} has all of its solutions real. In light of the computations described in Section 2.3, we felt this provided overwhelming evidence for the validity of Conjecture 2.1.

Our method was to solve the polynomial systems by elimination (see [10, §2] for a discussion of methods to solve systems of polynomial equations). We first choose distinct integral values of the parameters s_{i} and generate the resulting system of integral polynomial equations. Since we are performing an exact symbolic computation, we necessarily work with integral polynomials. Next, we compute an eliminant, a univariate polynomial g(x) with the property that its roots are the set of x-coordinates of solutions to our system. When g(x) has d = d_{m,p} roots (Schubert’s bound), there is a lexicographic Gröbner basis satisfying the Shape Lemma, since this system is zero-dimensional [11]. It follows that the solutions are rational functions (quotients of integral polynomials) of the roots of g(x). In some instances, the eliminant we calculated did not have d roots. For these we found a different eliminant with d roots. Lastly, we checked that these eliminants had only real roots.

Table 1 gives the number of instances we know have been checked. By Lemma 3.7(ii), there is a bijection between instances of (m, p) = (a, b) and (m, p) = (b, a). Table 1 also lists the running time to compute a degree reverse lexicographic Gröbner basis for the systems of Conjecture 1.1, and the size of that basis. This used Singular-1.2.1 [19] on a K6-2-300 processor with 256M running Linux. The instances reported in the last 3 columns are not due the the author. A more complete account is found in [42].

m, p  4,2  5,2  3,3 6,2  7,2 4,3 2,8
d_{m,p}  14  42  42 132 429 462 1430
# checked > 12, 000 1000 550 55  2  2  1
time (sec)  .04  1.42 1.50 78.6 8175 –  –
size  1.4K 12.8K 18.6K 202K 4.58M 32M –

Table 1. Instances checked

The computations of the last 2 columns stand out. The first is the case 8, 2 (also one instance each of 7, 2 and 4, 3) computed by Jan Verschelde [47] using his implementation of the SAGBI homotopy algorithm in [23]. Since the polynomial system of Conjecture 2.1 was ill-conditioned, he instead used the equivalent system of Conjecture 2.1′ (in Section 2.5 below), where the P_{i}(s) were the Chebyshev polynomials. These numerical calculations give approximate solutions whose condition numbers determine a neighborhood containing a solution. The solutions of this real system are stable under complex conjugation, so it sufficed to check that each neighbourhood and its complex conjugate were disjoint from all other neighborhoods. This computation took approximately 25 hours on a 166MHz Pentium II processor with 64M running Linux. These algorithms are ‘embarrassingly parallelizable’, and in principle they can be used to check far larger polynomial systems.

The second is the case of (m, p) = (3, 4) of Conjecture 1.1 (also all smaller cases with m ≤ p), computed by Faugère, Rouillier, and Zimmermann [15]. They first used FGB [13] to calculate a degree reverse lexicographic Gröbner basis for the system (2) for (m, p) = (3, 4) with s_{i} = i. This yielded a Gröbner basis of size 32M. They then computed a rational univariate representation [35] (a sophisticated substitute for an eliminant) in two ways. Once using a multi-modular implementation of the FGLM [14] algorithm and a second time using RS, an improvement of the RealSolving software [34] under development. The eliminant had degree 462 and size 3M, thus its general coefficient had 2,000 digits. Using an early implementation of Uspensky’s algorithm, they verified that all of its zeroes were real, proving Conjecture 1.1 for (m, p) = (3, 4). In the course of this calculation, they found it necessary to rewrite their software.

2.5. Equivalent systems. The extension of the map (4) to ℙ^{1}

γ : [t, s] ⟼ [t^{m+p−1}, st^{m+p−2}, …, s^{m+p−2}t, s^{m+p−1}]

is a parameterization of the standard real rational normal curve in ℙ^{m+p−1} and K(s) is the m-plane osculating this curve at the point γ[1, s]. In general, a parameterized real rational normal curve is a map γ : ℙ^{1} → ℙ^{m+p−1} of the form

[t, s] ⟼ [P_{1}(s, t), P_{2}(s, t), …, P_{m+p}(s, t)]

where P_{1}(s, t), …, P_{m+p}(s, t) form a basis for the space of real homogeneous polynomials in s, t of degree m + p − 1. All parameterized real rational normal curves are conjugate by a real projective transformation of ℙ^{m+p−1}. We deduce

Conjecture 2.1 (Geometric form) For all integers m, p > 1 and almost all choices of mp m-planes K_{1}, …, K_{mp} osculating a real rational normal curve at distinct real points, there are exactly d_{m,p} p-planes X satisfying

X ∩ K_{i} ≠ {0} for i = 1, …, mp

and all of these p-planes X are real.

Thus Conjecture 2.1 is equivalent to a conjecture concerning a much richer class of polynomial systems.

Conjecture 2.1′. Suppose m, p > 1 are integers and P_{1}(s), …, P_{m+p}(s) are a basis of the space of real polynomials of degree at most m + p − 1. Let K(s) be the m × (m + p) matrix of polynomials whose i, jth entry is P_{j}^{(i−1)}(s). Set

φ(s; X) := det [ \begin{matrix}K(s) \\ I_{p} X \end{matrix} ] .

Then, for almost all choices of distinct real numbers s_{1}, …, s_{mp}, the system

φ(s_{1}; X) = φ(s_{2}; X) = ⋯ = φ(s_{mp}; X) = 0

has exactly d_{m,p} solutions, and all of them are real.

The polynomial matrix K(s) of Conjecture 2.1′ differs from that of Conjecture 2.1 by right multiplication by an invertible (m + p) × (m + p)-matrix. Thus the resulting polynomial systems differ primarily by choice of local coordinates for the Grassmannian. In linear systems theory, two physical systems are output feedback-equivalent if their matrices of coprime factors [N(s) D(s)] differ in this manner [30].

We give an equivalent conjecture concerning a simpler system of polynomials with 2 fewer equations and unknowns. We may reparameterize the curve K(s) of Conjecture 2.1 and assume s_{mp−1} = 0 and s_{mp} = ∞. Observe that K(0) = [I_{p} 0] and K(∞) = [0 I_{p}]. The collection of all p-planes X satisfying

X ∩ [I_{p} 0] ≠ {0} and X ∩ [0 I_{p}] ≠ {0} (11)

is an irreducible rational variety of dimension mp − 2.

Let X be the set of all p × (m + p)-matrices X whose entries x_{i,j} satisfy:

x_{i,j} = 1 if j = i < p or (i, j) = (p, p + 1)
x_{i,j} = 0 if \{ \begin{matrix}i = 1 & and & j ≥ m \\ 1 < i < p & and & j < i or j > i + m \\ i = p & and & j ≤ p \end{matrix}

The remaining mp−2 entries are unconstrained and give coordinates for X. The row space of a matrix X is a p-plane X satisfying (11) and almost all such p-planes arise in this fashion. Thus X parameterizes a dense subset of the subvariety of p-planes X satisfying (11).

For example, if (m, p) = (4, 3), then X is the set of all matrices of the form:

[ \begin{matrix}1 & x_{12} & x_{13} & x_{14} & 0 & 0 & 0 \\ 0 & 1 & x_{23} & x_{24} & x_{25} & x_{26} & 0 \\ 0 & 0 & 0 & 1 & x_{35} & x_{36} & x_{37} \end{matrix} ] .

Since the 1, mth entry of a matrix in X vanishes,

det [ \begin{matrix}K(s) \\ X \end{matrix} ]

factors as s · ψ(s; X).

Conjecture 2.1′′. Let m, p > 1 be integers. Then, for almost all choices of non-zero real numbers s_{1}, …, s_{mp−2}, the system of equations

ψ(s_{1}; X) = ψ(s_{2}; X) = ⋯ = ψ(s_{mp−2}; X) = 0 (12)

is zero-dimensional with d_{m,p} solutions, and all of them are real.

The systems of Conjecture 2.1 and the variations given here are deficient: They have fewer solutions than standard combinatorial bounds. For example, the system (12) consists of mp − 2 equations of degree p, thus its Bézout number is p^{mp−2}. A better combinatorial bound is the normalized volume of the Newton polytope A_{m,p} of the polynomial ψ [26]. Table 2 compares these combinatorial bounds with d_{m,p}, for some values of m, p. The volumes of A_{m,p} were computed using PHC [46], a software package for performing general polyhedral homotopy continuation. Note the striking difference between the equivalent systems m, p and p, m.

\begin{matrix}m, p: & 2,2 & 3,2 & 4,2 & 5,2 & 6,2 & 7,2 & 8,2 & 2,3 & 3,3 & 4,3 & 2,4 & 3,4 \\ d_{m,p}: & 2 & 5 & 14 & 42 & 132 & 429 & 1430 & 5 & 42 & 462 & 14 & 462 \\ vol A_{m,p}: & 2 & 5 & 18 & 67 & 248 & 919 & 3426 & 5 & 130 & 3004 & 42 & 7156 \\ p^{mp−2}: & 4 & 16 & 64 & 256 & 1024 & 4096 & 16384 & 81 & 2187 & 59,049 & 4096 & 1048576 \end{matrix}

Table 2. Combinatorial bounds vs. d_{m,p}

2.6. Conjecture 2.1 for m = 2, p = 3.

Theorem 2.3. Conjecture 2.1 holds for (m, p) = (2, 3).

L. Gonzalez-Vega has also obtained this using resultants and Sturm-Habicht sequences.

Proof. We will prove the equivalent Conjecture 2.1′′. Let X := {x_{12}, x_{23}, x_{24}, x_{35}} be indeterminates. Set

ψ(s; X) = det [ \begin{matrix}1 & s & s^{2} & s^{3} & s^{4} \\ 0 & 1 & 2s & 3s^{2} & 4s^{3} \\ 1 & x_{12} & 0 & 0 & 0 \\ 0 & 1 & x_{23} & x_{24} & 0 \\ 0 & 0 & 0 & 1 & x_{35} \end{matrix} ]

We solve the system of polynomials

ψ(s; X) = ψ(t; X) = ψ(u; X) = ψ(v; X) = 0 (13)

by elimination.

The ideal ⟨ψ(s), ψ(t), ψ(u), ψ(v)⟩ in the ring ℚ(s, t, u, v)[x_{12}, x_{23}, x_{24}, x_{35}] has degree 5 = d_{2,3} and the lexicographic Gröbner basis with x_{12} < x_{23} < x_{24} < x_{35} contains the following univariate polynomial g, which is the universal eliminant for this family of systems:

x_{35}^{5}−4e_{1}x_{35}^{4}+(4e_{1}^{2}+6e_{2})x_{35}^{3}−(12e_{1}e_{2}+4e_{3})x_{35}^{2}+(9e_{2}^{2}+8e_{1}e_{3}−4e_{4})x_{35}−(12e_{2}e_{3}−8e_{1}e_{4})

Here e_{i} is the ith the elementary symmetric polynomial in s, t, u, v. We show that g has 5 distinct real roots for every choice of distinct parameters s, t, u, v. The discriminant Δ of g has degree 20 in the variables s, t, u, v and 711 terms: 9e_{3}^{4}e_{2}^{2}e_{1}^{4} − 54e_{3}^{4}e_{2}^{3}e_{1}^{2} + 81e_{3}^{4}e_{2}^{4} − 32e_{3}^{5}e_{1}^{5} + 204e_{3}^{5}e_{2}e_{1}^{3} − 324e_{3}^{5}e_{2}^{2}e_{1} − 108e_{3}^{6}e_{1}^{2} + 324e_{3}^{6}e_{2} + 81e_{4}^{2}e_{2}^{4}e_{1}^{4} − 486e_{4}^{2}e_{2}^{5}e_{1}^{2} + 729e_{4}^{2}e_{2}^{6} − 54e_{4}e_{3}^{2}e_{2}^{3}e_{1}^{4} + 324e_{4}e_{3}^{2}e_{2}^{4}e_{1}^{2} − 486e_{4}e_{3}^{2}e_{2}^{5} + 204e_{4}e_{3}^{3}e_{2}e_{1}^{5} − 1296e_{4}e_{3}^{3}e_{2}^{2}e_{1}^{3} + 2052e_{4}e_{3}^{3}e_{2}^{3}e_{1} − 8e_{4}e_{3}^{4}e_{1}^{4} + 738e_{4}e_{3}^{4}e_{2}e_{1}^{2} − 2106e_{4}e_{3}^{4}e_{2}^{2} − 108e_{4}e_{3}^{5}e_{1} − 324e_{4}^{2}e_{3}e_{2}^{2}e_{1}^{5} + 2052e_{4}^{2}e_{3}e_{2}^{3}e_{1}^{3} − 3240e_{4}^{2}e_{3}e_{2}^{4}e_{1} − 108e_{4}^{2}e_{3}^{2}e_{1}^{6} + 738e_{4}^{2}e_{3}^{2}e_{2}e_{1}^{4} − 2592e_{4}^{2}e_{3}^{2}e_{2}^{2}e_{1}^{2} + 3834e_{4}^{2}e_{3}^{2}e_{2}^{3} − 368e_{4}^{2}e_{3}^{3}e_{1}^{3} + 1800e_{4}^{2}e_{3}^{3}e_{2}e_{1} − 27e_{4}^{2}e_{3}^{4} + 324e_{4}^{3}e_{2}e_{1}^{6} − 2106e_{4}^{3}e_{2}^{2}e_{1}^{4} + 3834e_{4}^{3}e_{2}^{3}e_{1}^{2} − 972e_{4}^{3}e_{2}^{4} − 108e_{4}^{3}e_{3}e_{1}^{5} + 1800e_{4}^{3}e_{3}e_{2}e_{1}^{3} − 5544e_{4}^{3}e_{3}e_{2}^{2}e_{1} − 634e_{4}^{3}e_{3}^{2}e_{1}^{2} + 984e_{4}^{3}e_{3}^{2}e_{2} − 27e_{4}^{4}e_{1}^{4} + 984e_{4}^{4}e_{2}e_{1}^{2} + 432e_{4}^{4}e_{2}^{2} − 352e_{4}^{4}e_{3}e_{1} − 64e_{4}^{5}.

This vanishes when g has a double root. Thus the number of real roots of g is constant on each connected component (in ℝ^{4}) of the locus Δ ≠ 0. We show there is only one connected component, and so the number of real roots of g (and thus the original system) does not depend upon the choice of real parameters. Since the roots of g evaluated at (s, t, u, v) = (1, 2, 3, 4) are

8, 8 ± \sqrt{19}, 8 ± \sqrt{11}

it follows that there are always five real roots of g, and thus the system (13) has d_{2,3} = 5 real solutions whenever s, t, u, v are real and distinct.

We complete the proof. For w ∈ ℤ_{≥0}^{10}, consider the polynomial:

s^{w_{1}}t^{w_{2}}u^{w_{3}}v^{w_{4}}(s − t)^{w_{5}}(s − u)^{w_{6}}(s − v)^{w_{7}}(t − u)^{w_{8}}(t − v)^{w_{9}}(u − v)^{w_{10}} (14)

Let A_{w} be the primitive part of the symmetrization of this polynomial. Thus A_{w} is a sum of squares, none of which vanish on the locus where s, t, u, v are distinct. Then Δ is

\frac{1}{2}(7A_{2220222224} + 3A_{2222402204} + 6A_{4222022222} + 7A_{4220222222} + 2A_{4420022222}
+2A_{2222440022} + 2A_{0222443022} + A_{4420202222} + 2A_{4222420022} + A_{4220022422}
+A_{0222442202} + A_{2202024422} + 6A_{2222420024} + 10A_{4220022242} + 3A_{2222222222}).

Note that the term 7A_{4220222222} does not vanish when a single parameter is zero. Similarly, the term 3A_{2222402204} does not vanish when s = u and t = v (but u ≠ t). Thus the locus where Δ = 0 has dimension 1 and so its complement is connected.

We have a Maple program which performs the computations described and runs in ∼15 seconds on a K6-2-300 processor.

A positive semidefinite polynomial is a real polynomial that takes only nonnegative values. In the proof we showed Δ is positive semidefinite by exhibiting it as a sum of squares. Not all positive semidefinite polynomials are sums of squares of polynomials. There exist positive semidefinite polynomials of degree l in k variables which are not sums of squares of polynomials if min(k, l) > 2 and (k, l) ≠ (3, 4) [21]. For Δ, (k, l) = (4, 20).

The form of the squares we used (14) for the discriminant Δ, while motivated by the observation that no two parameters (0, s, t, u, v, ∞) should coincide, is justified by the observation that any real zero of Δ must also be a zero of all the squares, if Δ is a sum of squares. (See [8] for other applications of this idea.)

Each of the polynomials A_{w} is a sum of squares, the number given by the orbit of the symmetric group on its index w. Since 6 have trivial stabilizer, 7 are stabilized by a transposition, one by the dihedral group D_{8}, and one is invariant, there are 6 · 24 + 7 · 12 + 3 + 1 = 232 squares in all. This is not the best possible. Choi, Lam, and Reznick [9] show, for degree l homogeneous polynomials in k variables that are a sum of squares of polynomials, at most

Λ(k, l) := ⌊\frac{1}{2} (\sqrt{1 + 8\binom{k + l − 1}{l}} − 1)⌋

squares are needed. Note that Λ(4, 20) = 59.

3. Schubert conditions on a Grassmannian

3.1. The Schubert calculus on Grass(p, m + p). The enumerative problems of Section 2 are special cases of more general problems given by Schubert conditions on Grass(p, m + p). A Schubert condition on Grass(p, m + p) is an increasing sequence of integers

α : 1 ≤ α_{1} < α_{2} < ⋯ < α_{p} ≤ m + p.

Let \binom{[m+p]}{p} be the set of all such sequences. A Schubert variety Ω_{α}F_{}is given by a Schubert condition α and a complete flag K_{}in ℂ^{m+p}, a sequence of subspaces

K_{}: K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ K_{m+p} = ℂ^{m+p}

where dim K_{i} = i. Then the Schubert variety Ω_{α}F_{}is the set of all p-planes X satisfying

dim X ∩ K_{m+p+1−α_{i}} ≥ p + 1 − i (15)

for each i = 1, 2, …, p. This irreducible subvariety of Grass(p, m+p) has codimension |α| := ∑_{i}(α_{i} − i).

A sequence α^{}= α^{1}, …, α^{n} with α^{j} ∈ \binom{[m+p]}{p} with ∑_{j} |α^{j}| = mp is Schubert data for Grass(p, m + p). Given Schubert data α^{}and flags K^{1}, …, K^{n} in general position, there are finitely many (complex) p-planes X which lie in the intersection of the Schubert varieties Ω_{α^{j}}K^{j} for j = 1, …, n. The classical Schubert calculus [25] gives the following recipe for computing this number d = d(m, p; α^{}). Let h_{1}, …, h_{m} be indeterminates with degree(h_{i}) = i. For each integer sequence β_{1} < β_{2} < ⋯ < β_{r} define the following polynomial:

S_{β} := det(h_{β_{i}−j})_{1≤i,j≤r}. (16)

Here h_{0} := 1 and h_{i} := 0 if i < 0 or i > m. Let I be the ideal in Q[h_{1}, …, h_{m}] generated by those S_{β} with r = p + 1, 1 < β_{1}, and β_{p+1} ≤ m + p. The quotient ring A_{m,p} := Q[h_{1}, …, h_{m}]/I is isomorphic to the cohomology ring of Grass(p, m + p). It is Artinian with one-dimensional socle in degree mp. In the socle we have the relation

d · (h_{m})^{p} − S_{α^{1}}S_{α^{2}} ⋯ S_{α^{n}} ∈ I.

We can compute the number d by normal form reduction modulo any Gröbner basis for I.

If γ is a rational normal curve, then the flag of subspaces osculating γ at a point is the osculating flag to γ at that point.

Conjecture 3.1. Let m, p > 1 and α^{}be Schubert data for Grass(p, m + p). For almost all choices of flags K^{1}, …, K^{n} osculating a fixed rational normal curve at real points, there are exactly d(m, p, α^{}) p-planes X in the intersection of Schubert varieties

Ω_{α^{1}}K^{1}∩ Ω_{α^{2}}K^{2}∩ ⋯ ∩ Ω_{α^{n}}K^{n},

and each of these p-planes is real.

As with Conjecture 2.1, the intersection is zero-dimensional if the points of osculation are distinct [11], and there are no multiplicities for the important class of Pieri Schubert data, (described below) which includes the case of Conjecture 2.1.

If α_{i} = 1 + α_{i−1}, then condition (15) for i − 1 implies (15) for i. Thus only those conditions (15) with α_{i} − α_{i−1} > 1 (or α_{1} > 1) are essential, and so only the subspaces K_{m+p+1−α_{i}} corresponding to essential conditions need be specified in a flag. If α := (1, 2, …, p − 1, p + 1), then only the last condition is essential, thus the Schubert variety Ω_{α}K_{}consists of those X with dim X ∩ K_{m} ≥ 1. This shows Conjecture 2.1 is a special case of Conjecture 3.1.

3.2. Systems of polynomials. A complete flag K_{}is represented by a nonsingular matrix also written K_{}: The i-plane K_{i} is the row space of K_{i}, the first i rows of K_{}. The condition that dim X ∩ K_{m+p+1−α_{i}} ≥ p + 1 − i is given by

(m + p + 1 + i − α_{i})-minors of [ \begin{matrix}K_{m+p+1−α_{i}} \\ X \end{matrix} ] = 0.

The flag K_{}(s) osculating the rational normal curve γ with the parameterization (4) at γ(s) is represented by the (m + p) × (m + p)-matrix whose i, jth entry is \binom{j−i}{i−1}s^{j−i}.

Conjecture 3.1′. Let m, p > 1 and α^{}be Schubert data for Grass(p, m + p). For almost all n-tuples of distinct real numbers s_{1}, …, s_{n}, the system of polynomials

(m + p + 1 + i − α_{i}^{j})-minors of [ \begin{matrix}K_{m+p+1−α_{i}^{j}}(s_{j}) \\ I_{p} X \end{matrix} ] = 0

for i = 1, …, p and j = 1, …, n has d(m, p, α^{}) solutions, and each is real.

For any Schubert conditions α, β with α_{i} + β_{p+1−i} ≤ m + p for i = 1, …, p, let X_{α,β} be the collection of all p × (m + p)-matrices X whose entries x_{ij} satisfy

\begin{matrix}x_{i,α_{i}} = 1 & for i = 1, …, p \\ x_{i,j} = 0 & if j < α_{i} or j > m + p + 1 − β_{p+1−i} \end{matrix} (17)

If X ∈ X_{α,β}, then the row space of X is a p-plane in the intersection Ω_{α}K_{}(∞) ∩ Ω_{β}K_{}(0). In this way, X_{α,β} parameterizes a Zariski open subset of the set of all such p-planes. This parameterization can be used to obtain a system of equations simpler than, but equivalent to, the system of Conjecture 3.1′.

The map X_{α,β} → Grass(p, m + p) is not injective. For example, X_{123,134} consists of all 3 × 7-matrices of the form:

[ \begin{matrix}1 & x_{12} & x_{13} & x_{14} & 0 & 0 & 0 \\ 0 & 1 & x_{23} & x_{24} & x_{25} & 0 & 0 \\ 0 & 0 & 1 & x_{34} & x_{35} & x_{36} & x_{37} \end{matrix} ]

Let r_{1}, r_{2}, r_{3} be the rows of such a matrix. If x_{36} = x_{37} = 0, then for each a ∈ ℂ, the matrix with rows r_{1}, r_{2} + ar_{3}, r_{3} is in X_{123,134}, and these all have the same row space. Similarly, if x_{25} = 0, then the same is true of the matrices with rows r_{1} + ar_{2}, r_{2}, r_{3}.

Let X_{α,β}^{∘} ⊂ X_{α,β} be the set of those matrices whose entries further satisfy

For each i = 2, …, p, at least one x_{ij} ≠ 0, for j satisfying

β_{p+1−i} ≤ m + p + 1 − j < β_{p+2−i}. (18)

The map X_{α,β}^{∘} → Grass(p, m + p) is injective. For H_{123,134} this condition is that x_{25} ≠ 0 and (x_{36}, x_{37}) ≠ (0, 0).

3.3. Pieri Schubert conditions. If α ∈ \binom{[m+p]}{p} has α_{p−1} = p − 1 and α_{p} = p + a, then the Schubert variety Ω_{α}K_{}is

{X | X ∩ K_{m+1−a} ≠ {0}}.

We call such a Schubert condition a Pieri condition and denote it by J_{a}. Pieri Schubert data are Schubert data α^{1}, …, α^{n} were at most 2 of the conditions α^{i} are not Pieri conditions. These include the Schubert data of Conjecture 2.1.

Proposition 3.2 (Theorem 9.1 in [11]). If α^{}are Pieri Schubert data and the flags K^{1}, …, K^{n} osculate a rational normal curve at general points, then the intersection of Schubert varieties

Ω_{α^{1}}K^{1}∩ Ω_{α^{2}}K^{2}∩ ⋯ ∩ Ω_{α^{n}}K^{n},

is transverse. In particular, there are no multiplicities.

Here is the main theorem of this section.

Theorem 3.3. Let a, b > 1 and suppose that Conjecture 2.1 holds for this (m, p) = (a, b). Then Conjecture 3.1 holds for any Pieri Schubert data for Grass(p, m+p) with (p, m) ≤ (a, b) or (p, m) ≤ (b, a), in each coordinate.

Remark 3.4. If the conclusion of Proposition 3.2 held for all Schubert data, then the proof we give of Theorem 3.3 would imply its conclusion for all Schubert data as well.

Pieri conditions are special because of Pieri’s formula. For α, β ∈ \binom{[m+p]}{p} and a > 0, we write α <_{a} β if |α| + a = |β| and

α_{1} ≤ β_{1} < α_{2} ≤ β_{2} < ⋯ < α_{p} ≤ β_{p}.

Proposition 3.5 (Pieri’s Formula). Let J_{a} := 1 < 2 < ⋯ < p−1 < p+a ∈ \binom{[m+p]}{p}.

(i) In the cohomology ring ring A_{m,p} of Grass(p, m + p), S_{J_{a}} = h_{a} and

S_{α} · S_{J_{a}} = ∑_{α<_{a}β} S_{β}.

(ii) If K_{}(s) and K_{}(t) are flags osculating a rational normal curve at points s and

t, then

lim_{s→t} (Ω_{α}K_{}(t) ∩ Ω_{J_{a}}K_{}(s)) = ∑_{α<_{a}β} Ω_{β}K_{}(t).

Here, the limit is taken as cycles. By this we mean that the sum is the fundamental cycle of the limit of the schemes Ω_{α}K_{}(t) ∩ Ω_{J_{a}}K_{}(s) as s approaches t along the rational normal curve.

(iii) Suppose α^{}= α^{1}, J_{a}, α^{2}, …, α^{n} are Schubert data. Then

d(m, p; α^{}) = ∑_{α<_{a}β} d(m, p; β, α^{2}, …, α^{n}).

Statement (i) is the usual statement of Pieri’s formula [17, 22], Statement (ii) is Theorem 8.1 of [11], and Statement (iii) is a direct consequence of (i).

Proof. We deduce Theorem 3.3 from Lemma 3.7 below, which shows some simple dependencies between Conjecture 3.1 for different collections of Schubert data. Definition (15) implies that Ω_{β}K_{}⊂ Ω_{α}K_{}if and only if α ≤ β coordinatewise. In fact, Ω_{β}K_{}∩ Ω_{α}K_{}= Ω_{β∨α}K_{}, where β ∨ α is the coordinatewise maximum of α and β. We make some definitions needed for the statement of Lemma 3.7.

Definition 3.6. Let m, p > 1 be integers.

1. For α ∈ \binom{[m+p]}{p} define α^{⊥} ∈ \binom{[m+p]}{m} to be the increasing sequence obtained from the numbers {1, 2, …, m + p} \ {α_{1}, …, α_{p}}. Given Schubert data α^{}for Grass(p, m + p), set α^{⊥} to be (α^{1})^{⊥}, …, (α^{n})^{⊥}.

2. Suppose p > 2. For α ∈ \binom{[m+p−1]}{p−1} define α^{+} ∈ \binom{[m+p]}{p} to be 1 < 1 + α_{1} < ⋯ < 1 + α_{p−1}. Given Schubert data α^{}for Grass(p, m + p), set α^{+} to be (α^{1})^{+}, …, (α^{n})^{+}.

3. Let ⪯ be the partial order on Pieri Schubert data where we say that β^{}covers

α^{}= α^{1}, …, α^{n} if one of the following holds

β^{}= β, α^{3}, …, α^{n} with α^{2} = J_{a} and α^{1} <_{a} β
β^{}= α^{1}, …, α^{n−2}, β with α^{n−1} = J_{a} and α^{n} <_{a} β.

Lemma 3.7. Let m, p > 1 be integers.

(i) If α^{}is Schubert data for Grass(p, m+p), then α^{⊥} is Schubert data for Grass(m, m+ p). Moreover, Conjecture 3.1 holds for m, p, α^{}if and only if it holds for p, m, α^{⊥}.

(ii) Suppose p > 2 and let J_{m} := 1 < 2 < ⋯ < p−1 < p+m. If α^{}is Schubert data for Grass(p−1, m+p−1), then β^{}:= α^{+}, J_{m} is Schubert data for Grass(m, m+ p). Moreover, Conjecture 3.1 holds for m, p − 1, α^{}if and only if it holds for m, p, β^{}.

(iii) Let α^{}, β^{}be Pieri Schubert data for Grass(p, m + p) with α^{}⪯ β^{}. If Conjecture 3.1 holds for α^{}for Grass(p, m + p), then it holds for β^{}.

Proof of Theorem 3.3. First note that Conjecture 3.1 holds for Schubert data α^{}for Grass(p, m + p) if and only if it holds for any rearrangement of the data α^{}. Suppose Conjecture 2.1 holds for Grass(b, a + b). Let α^{}be Pieri Schubert data for Grass(p, m + p) where (m, p) ≤ (a, b) or (m, p) ≤ (b, a) coordinatewise. Since J_{1}^{⊥} = J_{1}, Conjecture 2.1 holds also for (m, p) = (b, a), by Lemma 3.7(i). Thus we may assume that (m, p) ≤ (a, b). By Lemma 3.7(ii), there exist Pieri Schubert data β^{}for Grass(b, a + b) such that Conjecture 3.1 holds for α^{}if and only if it holds for β^{}. Finally, Theorem 3.3 follows from (iii) by noting that the Schubert data of Conjecture 2.1, namely α^{1} = ⋯ = α^{ab} = J_{1}, is minimal among all Pieri Schubert data for Grass(b, a + b).

Proof of Lemma 3.7. For (i), fix a real inner inner product on ℂ^{m+p}. Then the map X ↦ X^{⊥} gives an isomorphism between Grass(p, m + p) and Grass(m, m + p). Given a flag K_{}and an increasing sequence α, let K^{⊥} be the flag of annihilators of the subspaces of K_{}. Then we have

X ∈ Ω_{α}K_{}⟺ X^{⊥} ∈ Ω_{α^{⊥}}K^{⊥}.

Furthermore, if K_{}(s) is the flag of subspaces osculating a rational normal curve γ at a point γ(s), then (K_{m+p−1}(s))^{⊥} is a rational normal curve with K^{⊥}(s) its osculating flag. Thus Conjecture 3.1 for Schubert data α^{}for Grass(p, m + p) is equivalent to Conjecture 3.1 for Schubert data α^{⊥} for Grass(m, m + p).

For (ii), let γ be the rational curve (4) with K_{}(s) as before. Then X ∈ Ω_{J_{m}}K_{}(∞) if and only if ⟨γ(∞)⟩ = K_{1}(∞) ⊂ X. Consider the projection π : ℂ^{m+p} ↠ ℂ^{m+p−1} from the last coordinate γ(∞). If X ∈ Ω_{J_{m}}K_{}(∞), then X′ := πX is a (p−1)-plane. This induces an isomorphism π : Ω_{J_{m}}K_{}(∞) ⟶^{∼} Grass(p−1, m+p−1). The inverse map is given by X′ ↦ K_{1}(∞) + X′.

The projection π ∘ γ is the standard rational normal curve γ′ in ℂ^{m+p−1}. Similarly, the flag K_{}′(s) osculating γ′ at γ′(s) is πK_{}(s). Note that if L is a linear subspace of ℂ^{m+p} with γ(∞) ∉ L, then dim X ∩ L = dim πX ∩ πL. In particular, if X ∈ Ω_{J_{m}}K_{}(∞), s ≠ ∞, and α ∈ \binom{[m+p−1]}{p−1}, then dim X′ ∩ K′_{(m+p−1)+1−α_{i}} ≥ (p − 1) + 1 − i if and only if dim X ∩ K_{m+p+1−(1+α_{i})} ≥ p + 1 − (i + 1). Thus we have

X ∈ Ω_{J_{m}}K_{}(∞) ∩ Ω_{α^{+}}K_{}(s) ⟺ X′ ∈ Ω_{α}K_{}′(s). (19)

In fact, this induces an isomorphism of schemes.

This gives a strong equivalence between enumerative problems: If α^{1}, …, α^{n} are in \binom{[m+p−1]}{p−1} and s_{1}, …, s_{n} any complex numbers, then the map π induces an isomorphism between the schemes

Ω_{J_{m}}K_{}(∞) ∩ ⋂_{i=1}^{n} Ω_{(α^{i})^{+}}K_{}(s_{i}) and ⋂_{i=1}^{n} Ω_{α^{i}}K_{}′(s_{i}).

Part (ii) follows by noting that any real reparameterization of the rational normal curve γ induces an isomorphism of polynomial systems, thus preserves real solutions. Hence given s_{0}, s_{1}, …, s_{n} ∈ ℙ_{ℝ}^{1}, there is an equivalent system with s_{0} = ∞.

It suffices to prove (iii) when β^{}covers α^{}in the partial order ⪯ defined on Pieri Schubert data. Suppose Conjecture 3.1 fails for β^{}and β^{}covers α^{}with α^{1} <_{a} β and α^{2} = J_{a} as in Definition 3.6 (iii). Then there exist distinct real numbers s_{1}, s_{3}, …, s_{n} such that

Ω_{β}K_{}(s_{1}) ∩ Ω_{α^{3}}K_{}(s_{3}) ∩ ⋯ ∩ Ω_{α^{n}}K_{}(s_{n}) (20)

is transverse with some complex p-planes in the intersection. We may assume without any loss that s_{1} = 0. Then there is an open subset O of the set of (n − 1)-tuples of real numbers s_{3}, …, s_{n} such that (20) is transverse and contains a complex p-plane X.

By the dimensional transversality results of [11], we may assume further that for β′ ∈ \binom{[m+p]}{p} and (s_{2}, …, s_{n}) ∈ O, the intersection

Ω_{β′}K_{}(0) ∩ ⋂_{i=3}^{n} Ω_{α^{i}}K_{}(s_{i})

has the expected dimension and is transverse if 0-dimensional. This is empty if |β′| > a + |α^{1}|, for dimension reasons. Thus

( ∑_{α<_{a}β′} Ω_{β′}K_{}(0)) ∩ ⋂_{i=3}^{n} Ω_{α^{i}}K_{}(s_{i})

is transverse for (s_{3}, …, s_{n}) ∈ O.

Fix (s_{3}, …, s_{n}) ∈ O. By Proposition 3.5(i), there is an ϵ > 0 such that for |t| < ϵ

Ω_{α}K_{}(0) ∩ Ω_{J_{a}}K_{}(t) ∩ ⋂_{i=3}^{n} Ω_{α^{i}}K_{}(s_{i})

is transverse. Here, when t = 0, replace Ω_{α}K_{}(0) ∩ Ω_{J_{a}}K_{}(t) by ∑_{α<_{a}β} Ω_{β′}K_{}(0). Since at t = 0 not all points in the intersection are real, the same holds for 0 < t < ϵ. But then Conjecture 3.1 fails for the Schubert data α^{}and completes the proof of Lemma 3.7.

3.4. An infinite family. We show that Conjecture 3.1 holds for an infinite family of nontrivial Schubert data.

Theorem 3.8. Conjecture 3.1 holds for any m with p = 2 and Pieri Schubert data where one condition is J_{m−1}.

Proof. By Lemma 3.7(iii), it suffices to show this for α^{1} = ⋯ = α^{m+1} = J_{1} and α^{m+1} = J_{m−1}. Geometrically, we are looking for the 2-planes which meet a 2-plane and m + 1 general m-planes nontrivially. We first show there are m such 2-planes. Let L = K_{2}(∞) = [0 I_{2}] and M = K_{m}(0) = [I_{m} 0], and let N_{i} = K_{m}(s_{i}), where s_{1}, …, s_{m} are distinct nonzero real numbers. For each one-dimensional subspace λ of L and each 1 ≤ i ≤ m, the composition

M ↪ L ⊕ M ≃ ℂ^{m+2} ↠ L ⊕ M/(λ + N_{i}) ≃ ℂ

defines a linear form ψ_{i,λ} on M. Each one-dimensional subspace μ of its kernel gives a 2-plane λ ⊕ μ containing λ and meeting both M and N_{i} nontrivially.

Thus if X is a 2-plane meeting L, M, and each N_{i} nontrivially, then H ∩ L = λ and H ∩ M = μ are lines with μ in the kernel of each form ψ_{i,λ}. Hence the forms are dependent. Similarly, if λ is a line in L such that the forms ψ_{i,λ} are dependent, then any line μ they collectively annihilate gives a 2-plane λ ⊕ μ meeting L, M, and each N_{i} nontrivially. It follows that the number of such 2-planes is the degree of the determinant of the forms ψ_{i,λ}, a polynomial in λ ∈ ℙ(L) ≃ ℙ^{1}. Since each form ψ_{i,λ} is a linear function of λ, the determinant has degree m, so there are m 2-planes X meeting L, M, and each N_{i} nontrivially.

We compute this determinant and show it has only real roots. Let λ = λ(x) be the span of the vector

(0, …, 0, 1, (m + 1)x).

Let the rational normal curve γ have the parameterization

γ : s ⟼ (1, −s, s^{2}, …, (−1)^{m+1}s^{m+1}).

Then K_{m}(s), the osculating m-plane to γ at γ(s), is the kernel of the matrix:

[ \begin{matrix}s^{m} & ms^{m−1} & … & \binom{m}{j}s^{m−j} & … & ms & 1 & 0 \\ 0 & s^{m} & … & \binom{m}{j−1}s^{m−j+1} & … & \binom{m}{2}s^{2} & ms & 1 \end{matrix} ] . (21)

If R_{j}(s) is the linear form given by the jth row of this matrix, then

((m + 1)x + ms_{i})R_{1}(s_{i}) − R_{2}(s_{i})

vanishes on λ(x) and its restriction to M gives the form ψ_{i,λ(x)}. This restriction is represented by the vector Λ(s_{i}, x) whose jth coordinate for j = 0, …, m − 1 is:

\binom{m+1}{j} ((m−j+1)xs_{i}^{m−j} + (m−j)s_{i}^{m−j+1}) .

We seek the determinant of the following matrix:

[ Λ(s_{1}, x) ; ⋮ ; Λ(s_{m}, x) ] .

This factors as A · B, where A is the bidiagonal m × (m + 1)-matrix

[ \begin{matrix}m & (m + 1)x & 0 \\ 0 & m^{2} − 1 & m(m + 1)x & 0 \\ ... & ... \\ 0 & \binom{m+1}{j}(m−j) & \binom{m+1}{j}(m−j+1)x & 0 \\ ... & ... \\ 0 & \binom{m+1}{2} & \binom{m+1}{2}2x \end{matrix} ]

and B is the (m + 1) × m-matrix whose i, jth entry is s_{j}^{m+2−i}. Numbering the rows of A and the columns of B from 0 to m, we see that

det(A(x) · B) = ∑_{i=0}^{m}(−1)^{i} det A_{i}(x) det B_{i},

where A_{i} is the matrix A with its ith column removed and B_{i} is the matrix B with its ith row removed. We find that

det(A_{i}) = m!(m + 1 − i)x^{m−i} ∏_{j=1}^{m} \binom{m+1}{j}
det B_{i} = e_{i}(s_{1}, …, s_{m})s_{1}s_{2} ⋯ s_{m} · ∏_{j<k}(s_{j} − s_{k})

and so det(A · B) is

m! ∏_{j<k}(s_{j} − s_{k}) ∏_{j=1}^{m} s_{j}\binom{m+1}{j} · (∑_{i=0}^{m}(−1)^{i}(m − i + 1)x^{m−i}e_{i}(s_{1}, …, s_{m})) .

Thus the coordinate x of the line λ satisfies the polynomial

P_{m}(s_{1}, …, s_{m}; x) := ∑_{i=0}^{m}(−1)^{i}(m − i + 1)x^{m−i}e_{i}(s_{1}, …, s_{m}).

Since we have e_{i}(s_{1}, …, s_{m}) = e_{i}(s_{1}, …, s_{m−1}) + s_{m}e_{i−1}(s_{1}, …, s_{m−1}), we see that

P_{m}(s_{1}, …, s_{m}; x) = (x − s_{m})P_{m−1}(s_{1}, …, s_{m−1}; x) + x ∏_{i=1}^{m−1}(x − s_{i}).

To complete the proof, we use induction to show that

(∗)  If 0 < s_{1} < ⋯ < s_{m}, then the roots r_{1}, …, r_{m} of P_{m} satisfy: 0 < r_{1} < s_{1} < r_{2} < s_{2} < ⋯ < r_{m} < s_{m}.

This suffices, if we can assume 0 < s_{1} < ⋯ < s_{m}. But we may assume this: Given a set of distinct real numbers s_{1}, …, s_{m}, s_{m+1}, s_{m+2}, we may assume s_{m+2} = ∞ and s_{m+1} < s_{1} < ⋯ < s_{m} and then apply the automorphism s ↦ s − s_{m+1} of ℙ^{1}(ℝ) which fixes ∞ = s_{m+2}.

The case m = 1 of (∗) holds as P_{1}(s_{1}; x) = 2x − s_{1}. Suppose P_{m−1} satisfies (∗). Then the roots of (x − s_{m})P_{m−1} are r_{1} < r_{2} < ⋯ < r_{m−1} < s_{m} and those of x ∏_{i=1}^{m−1}(x − s_{i}) are 0 < s_{1} < ⋯ < s_{m−1}. Moreover the leading coefficients of both polynomials are positive. The result follows by the Intermediate Value Theorem: If P (x) and Q(x) are polynomials of degree n with positive leading coefficients and real interlaced roots p_{i} of P and q_{i} of Q

p_{1} < q_{1} < p_{2} < q_{2} < ⋯ < p_{n} < q_{n},

then P (x) + Q(x) has real roots r_{i} satisfying p_{i} < r_{i} < q_{i}, for i = 1, …, n.

3.5. Computational evidence. We have proven Conjecture 3.1 in a number of cases besides those of Theorem 3.8. We also have done many computations along the lines of those in Section 2.4. To describe these, we use the following compact notation. If a Schubert condition α is repeated k times in some Schubert data, we abbreviate that by α^{k}. Thus, the conditions of Conjecture 2.1 are written as J_{1}^{mp}.

Theorem 3.9. Conjecture 3.1 holds for the following Schubert data.

(i) (m, p) = (4, 2), α^{}= J_{2}^{4}. Here, d(4, 2; J_{2}^{4}) = 3. (ii) (m, p) = (3, 3), α^{}= J_{2}^{4}, J_{1}. Here, d(3, 3; J_{2}^{4}, J_{1}) = 3. (iii) (m, p) = (3, 3), α^{}= (135)^{2}, J_{1}^{3}. Here, d(3, 3; (135)^{2}, J_{1}^{3}) = 6. (iv) (m, p) = (4, 3), α^{}= 135^{4}. Here, d(4, 3; 135^{4}) = 8.

Proof. We consider a polynomial system with parameters, give a universal eliminant, and show the eliminant has only real roots for distinct values of the parameters. We work in the local parameterization X_{α^{1},α^{2}} of Section 3.2.

(i) Let (m, p) = (4, 2) and α^{}= J_{2}^{4}. The equations are

maximal minors [ \begin{matrix}1 & s & s^{2} & s^{3} & s^{4} & s^{5} \\ 0 & 1 & 2s & 3s^{2} & 4s^{3} & 5s^{4} \\ 0 & 0 & 1 & 3s & 6s^{2} & 10s^{3} \\ 1 & x_{12} & x_{13} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_{25} & x_{26} \end{matrix} ] = 0

and the same equations with t replacing s. The ideal of these polynomials contains the following univariate polynomial g of degree 3 = d(4, 2, J_{2}^{4}).

25x_{12}^{3} − 25x_{12}^{2}(s + t) + x_{12}(19st + 6s^{2} + 6t^{2}) − 3(s^{2}t + st^{2})

whose discriminant has primitive part

9(s − t)^{6} + 23s^{2}t^{2}(s − t)^{2} + 9(s^{6} + t^{6}).

Since g(x_{12}; 1, 2) has roots

1, 1 ± \frac{1}{5}\sqrt{7},

we have shown that g always has real roots, when s and t are distinct.

(ii) Let m = p = 3 and α^{}= J_{2}^{4}, J_{1}. Here, X_{J_{2},J_{2}} consists of all matrics X of the form

[ \begin{matrix}1 & x_{12} & 0 & 0 & 0 & 0 \\ 0 & 1 & x_{23} & x_{24} & x_{25} & 0 \\ 0 & 0 & 0 & 0 & 1 & x_{36} \end{matrix} ]

and our equations are

det [ \begin{matrix}K_{3}(s) \\ X \end{matrix} ] = maximal minors [ \begin{matrix}K_{2}(t) \\ X \end{matrix} ] = 0

and the same equations with u replacing t. The ideal of these polynomials contains the following univariate polynomial g, here e_{1} = t + u and e_{2} = tu.

\begin{matrix}x_{36}^{3} − x_{36}(3s + 4e_{1}) + x_{36}(4e_{1}^{2} + 3e_{2} + 10se_{1}) − (6e_{1}e_{2} + 8se_{1}^{2} + se_{2}) \\ = (x_{36} − 2e_{1})(x_{36}^{2} − 2e_{1}x_{36} + 3e_{2}) − s(x_{36}^{2} − 10e_{1}x_{36} + 8e_{1}^{2} + e_{2}). \end{matrix}

These last two polynomials have roots

e_{1} ± \sqrt{e_{1}^{2} − 3e_{2}}, 2e_{1} and \frac{5}{3}e_{1} ± \frac{\sqrt{e_{1}^{2} − 3e_{2}}}{3} ,

which are interlaced. For example, if e_{1} > 0, then

e_{1} − \sqrt{e_{1}^{2} − 3e_{2}} < \frac{5}{3}e_{1} − \frac{\sqrt{e_{1}^{2} − 3e_{2}}}{3} < e_{1} + \sqrt{e_{1}^{2} − 3e_{2}} < \frac{5}{3}e_{1} + \frac{\sqrt{e_{1}^{2} − 3e_{2}}}{3} < 2e_{1}.

When s, t, u are distinct and different from 0, g always has 3 real roots, by the Intermediate Value Theorem. We could also note that the discriminant of g

\begin{matrix}s^{2}(t − u)^{4} + t^{4}(s − u)^{2} + u^{4}(s − t)^{2} + s^{2}t^{2}(s − t)^{2} + s^{2}u^{2}(s − u)^{2}+ \\ (s − t)^{2}(s − u)^{2}(t − u)^{2} + \frac{7}{2}(s^{4}(t − u)^{2} + t^{2}(s − u)^{4} + u^{2}(s − t)^{4} + t^{2}u^{2}(t − u)^{2}) \end{matrix}

is a sum of squares and g(x_{36}; 1, 2, 3) has (floating point) roots

4.736, 7.756, 10.508.

(iii) Let (m, p) = (3, 3) and α^{}= (135)^{2}, J_{1}^{3}. Here, X_{135,135} consists of all matrics X of the form

[ \begin{matrix}1 & x_{12} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & x_{24} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & x_{36} \end{matrix} ]

and our equations are

det [ \begin{matrix}K_{3}(s) \\ X \end{matrix} ] = det [ \begin{matrix}K_{3}(t) \\ X \end{matrix} ] = det [ \begin{matrix}K_{3}(u) \\ X \end{matrix} ] = 0.

We write the universal eliminant, g(x_{36}), in terms of the elementary symmetric polynomials in s, t, u:

\begin{matrix}9x_{36}^{6} − 48e_{1}x_{36}^{5} + (64e_{1}^{2} + 108e_{2})x_{36}^{4} − (288e_{1}e_{2} − 198e_{3})x_{36}^{3} \\ +(320e_{2}^{2} + 540e_{1}e_{3})x_{36}^{2} − 1200e_{2}e_{3}x_{36} + 1125e_{3}^{2}. \end{matrix}

Evaluating the parameters (s, t, u) at (1, 2, 3), we see that g(x_{36}; 1, 2, 3) has roots:

1.491, 1.683, 3.210, 5.630, 9.213, 10.773.

The discriminant of g is a sum of squares. The primitive part of the discriminant is

e_{3}^{4}(4e_{2}^{2}e_{1}^{2}−15e_{3}e_{1}^{3}−15e_{2}^{3}+63e_{3}e_{2}e_{1}−81e_{3}^{2})(256e_{2}^{2}e_{1}^{2}−768e_{3}e_{1}^{3}−768e_{2}^{3}+2592e_{3}e_{2}e_{1}−2187e_{3}^{2})^{2}.

The second factor is a sum of squares

\frac{7}{2}(s − t)^{2}(s − u)^{2}(t − u)^{2} + \frac{1}{2}s^{2}((t − u)^{4} + t^{2}(s − u)^{4} + u^{2}(s − t)^{4})

Interestingly, the last (squared) factor is itself a sum of squares:

\begin{matrix}112(s − t)^{2}(u^{4} + s^{2}t^{2}) + 112(t − u)^{2}(s^{4} + t^{2}u^{2}) + 112(u − s)^{2}(t^{4} + s^{2}u^{2})+ \\ 16(s − t)^{2}(s − u)^{2}(t − u)^{2} + 309s^{2}t^{2}u^{2} + 16(s^{4}(t^{2} + u^{2}) + t^{4}(s^{2} + u^{2}) + u^{4}(t^{2} + u^{2})). \end{matrix}

(iv) Let (m, p) = (4, 3) and α^{}= (135)^{4}. Here, X_{135,135} consists of all matrics X of the form

[ \begin{matrix}1 & x_{12} & x_{13} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & x_{24} & x_{25} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & x_{36} & x_{37} \end{matrix} ]

and our equations are

maximal minors [ \begin{matrix}K_{3}(s) \\ X \end{matrix} ] = maximal minors [ \begin{matrix}K_{5}(s) \\ X \end{matrix} ] = 0

and the same equations with t replacing s. In this case, the universal eliminant has 4 quadratic factors:

\begin{matrix}(36x_{12}^{2} − x_{12}(12t + 30s) + 6st + 5s^{2}), (36x_{12}^{2} − x_{12}(12s + 30t) + 6st + 5t^{2}), \\ (3x_{12}^{2} − 2x_{12}(s + t) + st), and (36x_{12}^{2} − 30x_{12}(s + t) + 5t^{2} + 14st + 5s^{2}). \end{matrix}

When s ≠ t and neither is zero, we see that each has 2 real roots.

Observe that in all 4 cases, the discriminant was a sum of squares and the eliminant has the correct number of real roots for distinct values of the parameters. Of particular note is that the system in (ii) was not symmetric in the parameters and the Schubert data of (iv) was not Pieri Schubert data.

Table 3 gives the number of instances of Conjecture 3.1 we have checked.

\begin{matrix}α^{}: & (J_{2})^{5} & (J_{2})^{6} & (J_{2})^{7} & (J_{2})^{6} & (J_{3})^{5} & (135)^{5} & (135)^{2}, (J_{1})^{6} \\ m, p: & 5,2 & 6,2 & 7,2 & 5,3 & 4,3 & 5,3 & 6,3 \\ d(m, p; α^{}): & 6 & 15 & 36 & 6 & 16 & 32 & 61 \\ # checked & 10,000 & 2821 & 504 & 10160 & 2002 & 400 & 294 \end{matrix}

Table 3. General Schubert data tested

4. Total positivity

Previous sections have dealt with Schubert conditions given by flags osculating a real rational normal curve. Recently, Shapiro and Shapiro have conjectured that a generalization of this choice involving totally positive real matrices would also give only real solutions. We describe that here, prove the first nontrivial instance, and present some computational evidence in support of this generalization.

A real upper triangular matrix g with 1’s on its diagonal is totally positive if every minor of g is positive, except those minors which vanish on all upper triangular matrices. Let T P be the set of all totally positive, a multiplicative semigroup. Define a partial order on real flags F_{}by F_{}< gF_{}if g ∈ T P.

Conjecture 4.1. For any m, p > 1, let α^{}be Schubert data for Grass(p, m + p). If F_{}^{1} < ⋯ < F_{}^{n} are real flags, then the Schubert varieties Ω_{α^{1}}F_{}^{1}, …, Ω_{α^{n}}F_{}^{n} intersect transversally, with all points of intersection real.

We will prove Conjecture 4.1 in the first nontrivial case of m = p = 2. First, we relate Conjecture 4.1 to Conjecture 3.1. Let K_{}(s) be the square matrix of size (m+p) whose i, jth entry is \binom{j−i}{i−1}s^{j−i} (cf. (1)). If s > 0, then K_{}(s) is totally positive and for any s, t we have K_{}(s) · K_{}(t) = K_{}(s + t). To see this, first recall that T P is generated as a semigroup by exp(E_{i,i+1}), where E_{i,i+1} is the elementary matrix whose only non-zero entry is in position i, i + 1 [27]. These assertions follow from the observation that

K_{}(s) = exp(sN),

where N is the nilpotent matrix whose only non-zero entries are (1, 2, …, m + p − 1) lying just above its main diagonal.

Theorem 3.3 holds in this new setting. For this, we alter the notion of Pieri Schubert data α^{}to Schubert data α^{1}, …, α^{n} where all except possibly α^{1} and α^{n} are Pieri conditions.

Theorem 4.2. Let a, b > 1 and suppose that Conjecture 4.1 holds for (m, p) = (a, b) and Schubert data α^{}= (J_{1})^{mp}. Then Conjecture 4.1 holds for any Pieri Schubert data for Grass(p, m + p) where (m, p) ≤ (a, b) or (b, a) coordinatewise.

Proof. The arguments used to prove Theorem 3.3 work here with minor adjustments.

We first remark that total positivity, and hence our order < on real flags, is defined with respect to a choice of ordered basis for ℝ^{m+p}. Suppose that e_{1}, …, e_{m+p} is the basis we used to define this order. Then F_{}< G_{}is and only if G_{}<′ F_{}, where <′ is defined with respect to the basis e_{1}, −e_{2}, e_{3}, −e_{4}, . . .. Similarly, if the basis e_{1}, …, e_{m+p} is orthonormal, then F_{}< G_{}if and only if F^{⊥} <′′ G^{⊥}, where <′′ is defined with respect to the basis in reverse order e_{m+p}, …, e_{2}, e_{1}. Thus

F^{1} < F^{2} < ⋯ < F^{n} ⟺ F^{n} <′ ⋯ <′ F^{2} <′ F^{1}

so that Conjecture 4.1 holds for Schubert data α^{}if and only if it holds for the data in reverse order. (This is the only rearrangment we used in the proof of Lemma 3.7.) Similarly, the analogue of Lemma 3.7(i) holds. For the analogue of Lemma 3.7(ii), permute the last two Schubert conditions, so that β^{}is still Pieri Schubert data, in our new, restricted definition.

Finally, in the proof of Lemma 3.7(iii), replace s_{3}, …, s_{n} in defining the set O by fixing F^{1} to be the standard flag represented by the matrix I_{m+p} and let O be the set of all

F^{1} < ⋯ < F^{n}

where the appropriate transversality conditions hold. Since T P is open, it follows that there exists ϵ > 0 and totally positive matrix M (which stabilizes F^{1}) such that if 0 < s < ϵ, then F^{1} < M · K_{}(s) · F^{1} < F^{2}. Then the same arguments used to prove Theorem 3.3 suffice. In particular, the analog of Proposition 3.2 also holds in this setting.

Totally positive matrices have a useful description. Let U be the group of real unipotent (upper triangular) matrices. Then T P is a connected component of the complement of a hypersurface H U defined by the vanishing of all minors consisting of the first i rows and last i columns [38]. This has a geometric description.

Associating a matrix to a flag as in Section 3.2, we may identify U with a Zariski open subset of the real flag manifold. Then the hypersurface H is the union of all positive codimension Schubert varieties defined by the flag determined by the identity matrix.

Given a matrix M ∈ U, the translate T P.M is a component of the complement of all Schubert varieties of positive codimension defined by the flag given by M. Similarly, given a totally positive matrix M, the set of upper triangular matrices N for which there exists a totally positive g with gN = M is the component of this complement containing the identity matrix.

Let F^{1} < ⋯ < F^{n} be real flags. Using a real automorphism of the flag manifold, we may assume that F^{1} = K_{}(0) = I_{m+p}. Then F^{2}, …, F^{n} ∈ T P, since they are all translates of the identity by totally positive matrices. Also, F^{1}, …, F^{n−1} are in the same component of the complement of all positive dimensional Schubert cells defined by F^{n}. If we now consider a real coordinate transformation fixing F^{1}, but with F^{n} becoming K_{}(∞), then this complement becomes T P, in these new coordinates.

Thus we may work in the local coordinates X := X_{α^{1},α^{n}}. We do this in our proof of the following theorem and in subsequent calculations.

Theorem 4.3. Conjecture 4.1 holds for m = p = 2 and Schubert data (J_{1})^{4}.

Proof. Let F, G ∈ T P be totally positive matrices and set H = G · F. When m = p = 2, X = X_{J_{1},J_{1}} is the set of matrices

[ \begin{matrix}1 & a & 0 & 0 \\ 0 & 0 & 1 & b \end{matrix} ] .

For a matrix L, let L_{ij} denote the 2 × 2-minor of L given by the first two rows and columns i and j. Then the equations for a 2-plane in X to meet the flags given by F and H are

f := F_{24} − bF_{23} − aF_{14} + abF_{13}
h := H_{24} − bH_{23} − aH_{14} + abH_{13}

The lexicographic Gröbner basis for this (with a < b) is

H_{13}f − F_{13}h = J_{14} − bJ_{24} − aJ_{34}
(H_{14} − bH_{13})f − (F_{14} − bF_{13})h = J_{13} − b(J_{23} + J_{14}) + b^{2}J_{24},

where J_{ij} is the ijth minor of the matrix:

[ \begin{matrix}F_{24} & F_{23} & F_{14} & F_{13} \\ H_{24} & H_{23} & H_{14} & H_{13} \end{matrix} ] .

We may write the the discriminant of the quadratic equation for b as follows

(J_{23} + J_{14})^{2} − 4J_{13}J_{24} = (L_{23} + L_{14})^{2} − 4L_{13}L_{24},

where L is the matrix

[ \begin{matrix}F_{13} & F_{14} & H_{13} & H_{14} \\ F_{23} & F_{24} & H_{23} & H_{24} \end{matrix} ] .

Thus we will have two real roots for our original system if and only if

Λ(B) := L_{13} − B(L_{23} + L_{14}) + B^{2}L_{24} = 0

has 2 real solutions. Painstaking calculations reveal that Λ(1) = −G_{12}G_{34} < 0. Since L_{24} = H_{13}H_{24} − H_{23}H_{14} = H_{12}H_{34} by the Plücker relations, we see that L_{24} > 0 and so Λ(B) = 0 will have 2 real solutions.

Table 4 shows the number of instances of Conjecture 4.1 that we have verified.

\begin{matrix}α^{} & (J_{1})^{6} & (J_{2})^{5} & (135)^{4} & (J_{1})^{8} & (J_{2})^{6} & (135)(136)(J_{1})^{5} \\ m, p & 3,2 & 5,2 & 4,3 & 4,2 & 6,2 & 4,3 \\ d & 5 & 6 & 8 & 14 & 15 & 25 \\ # checked & 12000 & 4000 & 4000 & 1500 & 300 & 150 \end{matrix}

Table 4. Instances checked

5. Further remarks

We present a counterexample to the original conjecture of Shapiro and Shapiro and close with a discussion of further questions.

5.1. A counterexample to the original conjecture. The original conjecture of Shapiro and Shapiro concerned the M-property for flag manifolds [37]. An algebraic set X defined over ℝ has the M-property if the sum of the ℤ/2ℤ-Betti numbers of X(ℝ) and of X(ℂ) are equal. Shapiro and Shapiro conjectured that an intersection of Schubert cells in a flag manifold has the M-property, if the cells are defined by flags osculating the rational normal curve at real points. When such an intersection is zero-dimensional all of its points are real. It is this consequence we have been studying.

While there is much evidence in support of this conjecture for zero dimensional intersections in a Grassmannian (Conjectures 2.1, 3.1, and 4.1), it does not hold for more general flag manifolds. In fact, we give a counter example in the simplest enumerative problem in a flag manifold that does not reduce to an enumerative problem in a Grassmannian.

Counterexample 5.1. Consider the manifold 𝔽(2, 3; 5) consisting of partial flags X ⊂ Y in ℂ^{5} with dim X = 2 and dim Y = 3. This manifold has dimension 8; the projection to Grass(2, 5) has fibre over a 2-plane X equal to ℙ(ℂ^{5}/X) ≃ ℙ^{2}. Given general 2-planes a, b, and c and general 3-planes A, B, and C, there are 4 flags X ⊂ Y which satisfy

X meets a, B, and C nontrivially dim Y ∩ A ≥ 2 and Y meets b and c nontrivially (22)

That this number is 4 may be verified using the Schubert calculus for a flag manifold [17] or the equations we give below.

Let K_{}(s) be the flag of subspaces osculating the standard rational normal curve. Set

\begin{matrix}a := K_{2}(4) & A := K_{3}(0) \\ b := K_{2}(1) & B := K_{3}(3) \\ c := K_{2}(−5) & C := K_{3}(−1) \end{matrix}

We claim that of the 4 flags X ⊂ Y satisfying (22) for this choice of a, b, c, A, B, C, 2 are real and 2 are complex.

We outline the computation. Choose local coordinates for 𝔽(2, 3; 5) as follows. Let Y be the row space of the 3 × 5-matrix

[ \begin{matrix}0 & 0 & 1 & x_{14} & x_{15} \\ 1 & 0 & x_{23} & x_{24} & x_{25} \\ 0 & 1 & x_{33} & x_{34} & x_{35} \end{matrix} ]

and X be the row space of its last 2 rows. We seek the solutions to the following overdetermined system of polynomials:

det [ \begin{matrix}K_{2}(1) \\ Y \end{matrix} ] = det [ \begin{matrix}K_{2}(−5) \\ Y \end{matrix} ] = det [ \begin{matrix}K_{3}(3) \\ X \end{matrix} ] = det [ \begin{matrix}K_{3}(−1) \\ X \end{matrix} ] =

maximal minors [ \begin{matrix}K_{2}(4) \\ X \end{matrix} ] = maximal minors [ \begin{matrix}K_{3}(0) \\ Y \end{matrix} ] = 0

These polynomials generate a zero-dimensional ideal containing the following univariate polynomial, which is part of a lexicographic Gröbner basis satisfying the Shape Lemma:

27063 − 117556x_{14} − 5952x_{14}^{2} − 10416x_{14}^{3} + 32400x_{14}^{4}.

This has roots

−.736 ± 1.30\sqrt{−1}, .227, 1.62.

Thus 2 of the flags are complex.

5.2. Further questions. While Counterexample 5.1 shows that we cannot guarantee all points of intersection real when the Schubert varieties are given by flags osculating a real rational normal curve, a number of questions remain (besides the resolution of the conjectures of the previous sections). There remains the original question of Fulton:

Question 1: Given Schubert data for a flag manifold, do there exist real flags in general position whose corresponding Schubert varieties have only real points of intersection?

In every case we know, this does happen. For instance, if we change the 3-plane B to K_{3}(2) in Counterexample 5.1, then all 4 solution flags are real. There is also the following result, showing this holds in infinitely many cases. Let 𝔽(2, n − 2; n) be the manifold of flags X ⊂ Y in ℂ^{n} where dim X = 2 and dim Y = n − 2. A Grassmannian Schubert condition is a Schubert condition on the flag X ⊂ Y which only imposes conditions on one of the subspaces. We likewise define Grassmannian Schubert data. For example, Counterexample 5.1 involves Grassmannian Schubert data.

Proposition 5.2 (Theorem 13 of [41]). Given any Grassmannian Schubert data for 𝔽(2, n − 2; n), there exist real flags whose corresponding Schubert varieties meet transversally with all points of intersection real.

The beauty of the conjectures of Shapiro and Shapiro is that they give a simple algorithm for selecting the flags defining the Schubert varieties.

Question 2: Can the choice of flags in Question 1 (or Proposition 5.2) be made effective? In particular, is there an algorithm for selecting these flags?

While computing the examples described here, we have made a number of observations which deserve further scrutiny. These concern eliminant polynomials in the ideals defining the intersections of Schubert varieties in the local coordinates we have been using.

Suppose we have Schubert data α^{}, and have chosen local coordinates either for the Grassmannian or are working in X_{α^{n},α^{n−1}}. Conjecture 3.1 or 4.1 may be formulated in terms of a parameterized system of polynomials with parameters either s_{1}, …, s_{n} in the case of Conjecture 3.1 or (n−1)-tuples of totally positive matrices (or in terms of some parameterization of T P [2]). For each of the coordinates, the ideal of this system contains a universal eliminant, which is the minimal univariate polynomial in that coordinate with coefficients rational functions in the parameters.

We ask the following questions about the eliminant.

Question 3: Does the universal eliminant have degree equal to the generic number of solutions? That is, do generic solutions satisfy the shape lemma?

Question 4: Let Δ be the discriminant of the polynomial system, a polynomial in the parameters which vanishes when there are solutions with multiplicities.

a) Is the locus Δ ≠ 0 connected?

b) In the case of Conjecture 3.1, where Δ is a polynomial in the parameters s_{1}, …, s_{n}, is Δ always a sum of squares of polynomials?

c) If so, are these polynomials monomials in the s_{i} and their differences (s_{i} − s_{j})? This would imply that the polynomial systems are always multiplicity-free for distinct real values of the parameters, and hence the stronger version of Theorem 3.3 mentioned in Remark 3.4.

The discriminants we have computed for instances of the conjectures for the Grassmannian (including the discriminant for system of Theorem 4.3) are always nonnegative when the parameters are distinct. For the case of Counterexample 5.1, we computed a discriminant for a simpler, but equivalent system, in the spirit of sections 2.5 and 3.2. This polynomial in parameters s_{1}, s_{2}, t_{1}, t_{2} is symmetric in the s’s and in the t’s separately (and in the transformation s_{i} ↔ t_{i}) and has degree 24. It has three factors, the first of degree 20 with 857 terms, and the square

(2s_{1}s_{2} + 2t_{1}t_{2} − (s_{1} + s_{2})(t_{1} + t_{2}))^{2}.

While this factor will not prevent the discriminant from being a sum of squares, this factor shows that there is a choice of distinct parameters for which the discriminant vanishes. Indeed, if we set s_{1} = 3, s_{2} = 6, t_{1} = 9, and t_{2} = 5, then this factor vanishes, and the resulting system has a root of multiplicity 2. This also explains why different values of the parameters in Counterexample 5.1 give different numbers of real and complex solutions.

Question 5: When the universal eliminant factors over ℤ, it reflects either some underlying geometry or some interesting arithmetic. More generally, one might ask about the Galois group of these enumerative problems [20], or the Galois group of the universal eliminant. For instance, is it the full symmetric group? That is not always the case, as the example of Theorem 3.9(iv) shows.

Question 6: In many cases with the substitution of s_{i} = i, the eliminant factors over the integers. This happens in Conjecture 1.1, Theorem 2.3, Theorem 3.9(i) and (iv), and in other cases. Table 5 lists the degrees of the factors in the case of Conjecture 1.1. Why does this choice of s_{i} = i induce a factorization? Is there any

m, p 3,2 4,2 5,2  6,2 7,2 3,3  3,4
d_{m,p}  5 14 42  132 429 42  462
Factors 2,3 6,8 10,32 20,112  6,36 16,30,416

Table 5. Factorization of the eliminant

special geometry or interesting arithmetic here? If 2 parameters are allowed to come together, then the resulting ideal factors in a way respecting the product of Schubert classes, by the Corollary to Theorem 1 in [12]. From the Schubert calculus, we would expect factors of 9 and 5 for (m, p) = (2, 4), 14 and 28 for (m, p) = (2, 5), and 21 and 21 for (m, p) = (3, 3), but these do not appear in Table 5.

Acknowledgements

I thank Boris Shapiro for sharing these conjectures and Joachim Rosenthal with whom I found the first strong evidence for their validity [33]. Bernd Sturmfels helped me learn computer algebra and suggested using discriminants. This work also benefitted from discussions with David Eisenbud, Ioannis Emiris, Jean-Charles Faugère, Bill Fulton, Birk Huber, Bernard Mourrain, Bruce Reznick, Fabrice Rouillier, Jan Verschelde, and Paul Zimmermann. Lastly, this would not be possible were it not for the authors of the following software packages: Linux, Maple, Singular, Macaulay2, MuPAD, FGB, RealSolving, PHC, and also my brothers Larry Sottile and Joe Sottile who helped me build the computer which performed most of these experiments.

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Frank Sottile, Department of Mathematics, University of Wisconsin, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388, USA

  E-mail address: sottile@math.wisc.edu
  URL: http://www.math.wisc.edu/~sottile