Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

Discrete & Computational Geometry, Dec 2010

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov, and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile’s monotonicity conditions are not satisfied.

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Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

Discrete Comput Geom Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties Monique Azar 0 1 Andrei Gabrielov 0 1 0 A. Gabrielov ( ) Department of Mathematics, Purdue University , West Lafayette, IN 47907 , USA 1 M. Azar Department of Mathematics, American University of Beirut , Beirut , Lebanon The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov, and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile's monotonicity conditions are not satisfied. Schubert calculus; Wronski map; Real enumerative geometry; Rational functions - 1 Introduction Schubert Calculus is a recipe for counting geometric objects subject to certain incidence relations [ 17 ]. For example: Problem 1.1 Given 2d − 2 lines in general position in CPd , how many codimension 2 subspaces of CPd meet all 2d − 2 lines? The answer is ud = d1 2dd−−12 , the d th Catalan number. Schubert Calculus was based on the semi-empirical principle of conservation of number. Its rigorous foundation, subject of Hilbert’s 15th problem, was established through the development The second author was supported by the NSF grant DMS-0801050. of intersection theory. From the point of view of intersection theory, enumerative problems such as Problem 1.1 are solved by counting intersection multiplicities of Schubert varieties in the Grassmannian (see, e.g., [ 1 ]). Over the reals, the principle of conservation of number fails, and solving enumerative problems becomes considerably more complicated. In [ 7 ], W. Fulton says “The question of how many solutions of real equations can be real is still very much open, particularly for enumerative problems.” For example, suppose that all 2d − 2 lines in Problem 1.1 are real. How many codimension 2 subspaces that meet all these lines are real? Sottile [ 23 ] proved that all of them can be real for some choice of the 2d − 2 lines. Boris and Michael Shapiro conjectured that all these subspaces are real whenever the given lines are tangent to the rational normal curve γ (t ) = [1 : t : t 2 : · · · : t d ] at distinct real points. Eremenko and Gabrielov [ 4 ] (see also [ 5 ]) proved the following equivalent theorem. Theorem 1.2 If the critical points of a rational function f are all real, then f is equivalent to a real rational function, i.e., there exists a fractional linear transformation l such that l ◦ f is a real rational function. A more general version of the B. and M. Shapiro conjecture, proved in [ 12 ],1 claims a similar result for higher-dimensional subspaces. Theorem 1.3 Let P1, P2, . . . , Pk(d−k+1) be (k − 1)-dimensional planes in CPd osculating the rational normal curve at distinct real points. Any (d − k)-dimensional plane that meets all Pi , 1 ≤ i ≤ k(d − k + 1), must be real. B. and M. Shapiro suggested an extension of their conjecture to flag varieties, replacing osculating planes by osculating flags. A special case of that conjecture would imply that all solutions to the following problem are real. Problem 1.4 Let y1, y2, . . . , y2d−3, r, s be distinct points in R. For 1 ≤ i ≤ 2d − 3, let Ti be the line tangent to γ at γ (yi ), and let T2d−2 be the line through the points γ (r) and γ (s). Among all the codimension 2 subspaces of CPd that meet all the lines T1, T2, . . . , T2d−2, how many are real? Sottile [ 20 ] found examples when the B. and M. Shapiro conjecture for flag varieties fails, in particular, Problem 1.4 does have nonreal solutions. Computer experiments suggested that the conjecture might hold whenever a certain monotonicity condition is met [ 16 ]. For the case of Problem 1.4, Sottile’s monotonicity condition simply means that the interval I with endpoints r and s contains either all or none of the points y1, y2, . . . , y2d−3. A proof of this special case was given in [ 6 ]. For a complete survey of the B. and M. Shapiro conjecture and its modifications, see [ 21 ] or [ 22 ]. 1This result dates back to 2005. In this paper we study Problem 1.4 when the monotonicity condition does not hold, i.e., when the interval I contains k of the points y1, y2, . . . , y2d−3 for some k with 1 ≤ k < 2d − 3. We give an algorithm to compute a (nonstrict) lower bound2 for the number of real subspaces for any pair of integers d and k with 1 ≤ k < 2d − 3. We do this by giving a lower bound for the number of solutions to the following equivalent problem. Problem 1.5 Let y1, y2, . . . , y2d−3, r, s be distinct points in R and suppose that the interval I having endpoints r and s contains k of the points y1, y2, . . . , y2d−3 for some k ∈ {1, 2, . . . , 2d − 4}. How many equivalence classes of real rational functions f of degree d having critical points at y1, y2, . . . , y2d−3 and satisfying f (r) = f (s) are there? Remark 1.6 The answer to Problem 1.5 does not change if we replace k by 2d − 3 − k. Lower bounds have been computed for several other real enumerative problems. Eremenko and Gabrielov [ 3 ] showed that the number of real solutions to certain systems of real polynomial equations is bounded below by the topological degree of the corresponding Wronski map and computed this degree. Soprunova and Sottile [ 19 ] extended the definition of the real Wronski map to certain sparse polynomial systems associated with partially ordered sets, computed its degree, and derived the corresponding lower bounds on the number of real solutions. Degtyarev and Kharlamov [ 2 ] showed that there are at least 8 real rational cubics passing through 8 general real points in the plane. Welschinger [ 26 ] discovered an invariant that gives a lower bound for the number of real rational curves of degree d passing through 3d − 1 real points. Itenberg, Kharlamov, and Shustin [ 10 ] showed that this invariant is at least d!/2 using Mikhalkin’s tropical correspondence theorem [ 11 ]. Welschinger discovered other invariants for similar enumerative problems (see [ 25 ] and the references there), some of which have been computed by Pandharipande, Solomon, and Walcher [ 13 ]. Solomon [ 18 ] showed how the Welschinger invariant can be interpreted as the degree of a map. An excellent review of these results can be found in [ 21, 22 ]. While the previous results are based on topological degree invariants, our method uses a different line of reasoning that does not seem to be associated with a topological degree. The lower bound for the number of real solutions of Problem 1.5 is obtained by studying a one-parameter family of real rational functions with critical points y1, . . . , y2d−3, θ and the dependence on θ of the increments on [r, s] of the functions in that family. 2 The Wronski Map and One-Parameter Families of Rational Functions In this section, we represent the points of the Grassmannian G of two-dimensional planes in the space of polynomials of degree d by pencils (p, q) of linearly independent polynomials, define the Wronski map W : G −→ CP2d−2, and prove that W 2The lower bound is strict for d = 4, 5, and 6. is not ramified over the space Q of polynomials with all real roots of multiplicity at most 2. We study the properties of a one-parameter family ft = pt /qt of rational functions obtained by lifting to G a path t → wθ(t) in Q. Here θ (t ) is a path in RP1, and wθ is a polynomial with roots y1, . . . , y2d−3, θ . Our main tool is a net of a real rational function with real critical points (see [ 4, 5 ]). Let G be the Grassmannian of two-dimensional planes in the space of complex polynomials of degree at most d . An element of G can be defined by a pencil L = L(p, q) = {αp + βq}, where p and q are two linearly independent polynomials of degree at most d , and [α : β] ∈ CP1. If L ∈ GR, the Grassmannian of real two-dimensional planes, we always assume that p and q are real and call L a real pencil. The Wronskian of (p, q), given by W (p, q) = p q − q p, is a nonzero polynomial of degree at most 2d − 2. If the degree of W (p, q) is 2d − 2 − k for some k > 0, we say that W (p, q) has a root at infinity of multiplicity k. If (p1, q1) is another basis for L, then W (p1, q1) differs from W (p, q) by a nonzero multiplicative constant. For a nonzero polynomial r = ai zi of degree at most 2d − 2, let h(r) = [a0 : a1 : . . . : a2d−2]. Definition 2.1 The Wronski map W : G −→ h(W (p, q)), where (p, q) is a basis for L. CP2d−2 is defined by W(L) = The map W is well defined and finite. Its degree is u(d), the d th Catalan number. This was originally computed by Schubert in 1886. A proof can be found in [ 8 ] or [ 9 ]. Definition 2.2 Let Q = h(Q0), where Q0 is the set of all polynomials r satisfying (i) 2d − 4 ≤ deg(r) ≤ 2d − 2, (ii) all roots of r belong to RP1, and (iii) all roots of r have multiplicity at most 2. In Theorem 2.13, we shall show that W is unramified over Q and hence, W|W−1(Q) is a covering map. Thus, for any path ρ in Q with initial point ρ0, given L0 ∈ G with W(L0) = ρ0, ρ can be lifted in a unique way to a path in G starting at L0. Definition 2.3 Let D = {y1, y2, . . . , y2d−3} be a set of points in R. For y ∈ R, let ωy = (x − y1)(x − y2) . . . (x − y2d−3)(x − y), and let ω∞ = (x − y1)(x − y2) . . . (x − y2d−3). The map RP1 −→ CP2d−2 given by y −→ h(wy ) is continuous. In fact, it extends to an algebraic map CP1 −→ CP2d−2. Let I be an interval in R, and let t0 ∈ I. A path θ : I −→ RP1 induces a path ρ : I −→ Q given by ρ(t ) = h(ωθ(t)). Given L0 ∈ G with W(L0) = h(ωθ(t0)), ρ can be lifted to a unique path χ in G satisfying χ (t0) = L0. It was shown in [ 4 ] that W−1(RP2d−2) ⊂ GR. This implies that χ (I) is contained in GR. Lemma 2.4 Let Rd be the space of real rational functions of degree at most d all of whose critical points are real. If θ is analytic, all the lifts defined above are analytic, and polynomials pt and qt with coefficients analytic in t can be selected to form the bases of χ (t ), t ∈ I. Let ft be the real rational function pt /qt viewed as a holomorphic function from CP1 to CP1. The path η : I −→ Rd given by t −→ ft has the following properties. (i) The Wronskian W (pt , qt ) is equal to ρ(t ). (ii) If x ∈ CP1 \ D, then the map t −φ→x ft (x) is continuous. φx (iii) For each x ∈ D, the map t −→ ft (x) is continuous at all values t for which pt and qt do not have a common root at x. Proof The first property follows from the construction of ft . To prove (ii) and (iii), first observe that if x ∈ CP1 \ D, then, for any t ∈ I, pt (x) and qt (x) cannot both be zero. Let (t0, x) ∈ I × CP1 be such that at least one of pt0 (x) and qt0 (x) is not equal to 0. This implies that there exists a neighborhood N of t0 such that either ft (x) or 1/ft (x) is a continuous complex-valued function on N and hence, ft (x) : N −→ CP1 is continuous. For pairs (t, x) such that pt (x) = qt (x) = 0, we shall see in the following lemma that for small > 0, ft+ is monotone on (x, x + ), and the image of (x, x + ) covers RP1 except for an interval of length O( ). Lemma 2.5 Suppose that y1 = 0 and that {(p , q ) : ≥ 0} is a family of ordered pairs of linearly independent real polynomials of degree at most d satisfying: (i) the coefficients of p and q are analytic in , (ii) the roots of w = W (p , q ) are y1 = 0, y2, . . . , y2d−3, , (iii) p0 and q0 have a common root at x = 0, and (iv) f0(0) = 0. Then for small > 0, f (0) = O( ), f ( ) = O( ), and f has a pole in (0, ). Moreover, f : [0, ] −→ RP1 is one-to-one, and its image is RP1 \ (α, β), where (α, β) is an open interval of size O( ) with endpoints at f (0) and f ( ). Proof By our assumption, p0 and q0 have a common root at x = 0, and f0(0) = 0, so there exist k1 = 0 and k2 = 0 such that p0 = k1x2 + O(x3) and q0 = k2x + O(x2). The requirement that f has a fixed critical point at x = 0 for all > 0 implies that p (x) = k1(x2 + a x + ab 2) + O(x3, x2, 2x, 3) and q (x) = k2(x + b ) + O(x2, x, 2). Therefore, w (x) = k1k2(x2 + 2b x) + O(x3, x2, 2x, 3). In particular, w ( ) = (1 + 2b)k1k2 2 + O( 3). Since for all > 0, f has a critical point at x = , we should have b = − 21 . Hence, q has a root at x = 21 + O( 2) between the critical points 0 and of f , and p (x ) = 0. Since p (0) = O( 2) and p ( ) = O( 2), we have f (0) = O( ) and f ( ) = O( ). For small enough > 0, the interval (0, ) does not contain any points of D. If f < 0 on (0, ), then when x moves from 0 to , the value of f (x) decreases from O( ), tends to −∞ as x approaches x , and returns from ∞ to O( ) at x = . If f > 0 on (0, ), then when x moves from 0 to , the value of f (x) increases from O( ), tends to ∞ as x approaches x , and returns from −∞ to O( ) at x = . Suppose t −→ gt = pt∗/qt∗ is another path satisfying L(pt∗, qt∗) = χ (t ) = L(pt , qt ). There exists a real fractional linear transformation ht = (at x + bt )/(ct x + dt ) such that gt = ht ◦ ft . In particular, for each t , ft and gt have the same critical points. Observe that the continuity of pt , qt , pt∗, qt∗ implies that the sign of at dt − bt ct does not change as t varies in I. Post-composition with a real fractional linear transformation defines an equivalence relation on Rd . This proves the following. Theorem 2.6 With the above notation, let ft0 ∈ Rd be a function whose Wronskian is equal to ρ(t0). Let Rd0 be the family of equivalence classes of functions in Rd of degree exactly d , and let [ft0 ] ∈ Rd0 be the equivalence class of ft0 . There exists a unique map η : I −→ Rd0 with η(t0) = [ft0 ] that projects to ρ. The set Rd0 is in one-to-one correspondence with the set of nondegenerate real pencils. A pencil L(p, q) ∈ G is nondegenerate if deg(p/q) = d , in other words, if max(deg p, deg q) = d and p and q have no common zeros. Otherwise, L is said to be degenerate. Notation 2.7 We shall identify RP1 with the unit circle S1 via a fractional linear transformation ϑ : CP1 −→ CP1 mapping RP1 to S1, 0 to 1, and preserving orientation. A real rational function f will be identified with the function f = ϑ ◦ f ◦ ϑ −1. The critical points of f are the images of the critical points of f under ϑ . In particular, all critical points of f belong to the unit circle. Moreover, for any two points a and b in RP1, f (a) = f (b) if and only if f (ϑ (a)) = f (ϑ (b)). Let Sd = {f = ϑ ◦ f ◦ ϑ −1 : f ∈ Rd }. We shall write f instead of f and specify whether we are considering f as a function in Rd or Sd . With this identification, any construction or proof given for Rd also applies to Sd and vice versa. Definition 2.8 Let L = L(p, q) be a real pencil whose Wronskian has 2d − 2 real roots counted with multiplicity, and let f = p/q. The set Γ = f −1(RP1) is called the net of f [ 4 ]. Alternatively, let f = ϑ ◦ f ◦ ϑ −1. Then Γ = f −1(S1). Observe that Γ is independent of the choice of basis (p, q), so we can also refer to Γ as the net of L. Unless explicitly stated otherwise, we shall consider Γ to be defined as f −1(S1) rather than f −1(RP1). Clearly, S1 ⊂ Γ , and Γ is invariant under the reflection s(z) = 1/z with respect to the unit circle. The net Γ defines a cell decomposition of CP1 as follows: (1) the vertices (0-cells) of the cell decomposition correspond to the zeros of W (p, q), (2) the edges (1-cells) are the components of Γ \ V , where V is the set of vertices, (3) the faces (2-cells) are the components of C \ Γ . This cell decomposition has the following properties. The closure of each cell is homeomorphic to a closed ball of the corresponding dimension. If W (p, q) has distinct real roots, then each vertex of Γ has degree 4. If W (p, q) has multiple roots, we call Γ a degenerate net. Remark 2.9 Let L(p, q) be a real pencil whose net Γ is degenerate and whose Wronskian has only simple or double roots, all real. A vertex of Γ of degree 2 corresponds to a point x such that p(x) = q(x) = 0, while a vertex of degree 6 corresponds to a double root x of W (p, q) such that at least one of p(x) and q(x) is not 0. Definition 2.10 The edges of Γ that lie inside the unit disc are called chords or interior edges of Γ . Since Γ is symmetric with respect to S1, it suffices to consider its chords. Given a vertex v, the pair γ = (Γ , v) is called the net of f with respect to v, or simply the net of f if v is clear. The vertex v is called the distinguished vertex of γ . The standard orientation of S1 induces a cyclic order ≺ on the vertices of Γ . We shall label these vertices v1, v2, . . . , v2d−2 so that v1 ≺ v2 ≺ · · · ≺ v2d−2 ≺ v1, and we shall set v2d−1 = v1. For a net γ = (Γ , v1), we define a linear order on the vertices by v1 < v2 < · · · < v2d−2. Two nets (Γ , v1) and (Γ , v1) are said to be equivalent if there exists a homeomorphism CP1 −→ CP1 mapping Γ to Γ , v1 to v1, preserving orientation of both CP1 and S1, and commuting with the reflection s. By abuse of notation, we shall often refer to an equivalence class of nets by one of its representatives. Definition 2.11 For a net γ = (Γ , v), let Shift(γ ) = (Γ , v ) where v is the predecessor of v under ≺. Remark 2.12 Without loss of generality, we may assume that the vertices of Γ are equally spaced on the circle. The net Shift(γ ) is equivalent to the net (Γ , v) where the ordered vertex set of Γ coincides with that of Γ , and the chords of Γ are obtained by rotating the chords of Γ by π/(d − 1) counterclockwise. Theorem 2.13 The Wronski map W : G −→ CP2d−2 is unramified over the space Q of all polynomials with real roots of multiplicity at most 2. Proof Let π0 ∈ Q, L(p0, q0) ∈ W−1(π0), and f0 = p0/q0. The function f0 can be chosen real [ 4 ]. From the definition of Q, the common roots r1, . . . , rm of p0 and q0 (if any) must be simple. Suppose that W is ramified at L(p0, q0), with ramification index ν. Let πj be a sequence of polynomials with distinct real roots converging to π0 as j → ∞. Since the polynomials with distinct real roots form an open set in the space of all real polynomials, we can assume that W is not ramified over πj for all j ≥ 1. Then W−1(πj ) contains ν distinct points L(pj,k, qj,k) of the Grassmannian converging to L(p0, q0) as j → ∞. Theorem 1.2 implies that all pencils L(pj,k, qj,k) are real. With the proper normalization, we can assume that pj,k and qj,k are real polynomials and that (pj,k, qj,k) → (p0, q0) as j → ∞. Then the rational functions fj,k = pj,k/qj,k are real, with all real critical points, converge to f0 as j → ∞ uniformly on every compact set not containing the points r1, . . . , rm, and their nets converge to the (degenerate) net of f0. Since the roots of π0 are at most double, the net of f0 may have vertices of degree either 6 or 2. In both cases, the net of fj,k is uniquely determined by the net of f0 for large enough j , independent of k. It follows from [ 4 ] that, for large enough j , pencils L(pj,k, qj,k) do not depend on k. Hence, ν = 1, and W is not ramified over π0. 3 Properties of the Nets of Functions ft In this section, we study dependence on the parameter t of the net γt of the rational function ft defined in Sect. 2. In particular, we show that the net γt is shifted counterclockwise when θ (t ) passes its first vertex y1. We describe also degenerate nets corresponding to polynomials wθ with double roots. Definition 3.1 A 2 × (d − 1) Young tableau is the distribution of the integers 1, 2, . . . , 2d − 2 on a 2 × (d − 1) rectangular array such that every integer is greater than the one above it and the one to its left, if any. Equivalence classes of nets with 2d − 2 vertices can be identified with 2 × (d − 1) Young tableaux: to a net γ = (Γ , v1) with the ordered vertex set {v1, v2, . . . , v2d−2}, we associate the tableau with an integer i belonging to the first row if and only if vi is connected by a chord to a vertex vj with j > i. In particular, the number of equivalence classes of nets having 2d − 2 vertices is the d th Catalan number ud = 1 2d−2 (see [ 24 ]). d d−1 Example 3.2 For d = 4, there are u4 = 5 equivalence classes of nets with 6 vertices. These equivalence classes (with the distinguished vertex v1 = 1) and their respective Young tableaux are shown in Fig. 1. Let D = {y1, y2, . . . , y2d−3} be an ordered set of distinct points in S1 all different from 1. Let θ : [0, 2d − 2] −→ S1 be the path given by θ (t ) = e2πit . By Lemma 2.4 and Notation 2.7, there exists a family {ft ∈ Sd : t ∈ [0, 2d − 2]} with the following properties: (i) for each t , the critical points of ft counted with multiplicity are y1, y2, . . . , y2d−3, and θ (t ); (ii) given (t0, z0) ∈ [0, 2d − 2] × CP1, ft (z0) as a function of t is continuous at t0 whenever deg ft0 = d . In particular, for any z0 ∈/ D, ft (z0) is a continuous function of t . For t ∈ [0, 2d − 2], let γt = (Γt , y1) be the net of ft with respect to y1. Let θ −1(D) = {t1, t2, . . . , tκ−1} with 0 < t1 < t2 < · · · < tκ−1 < 2d − 2, and let t0 = 0 and tκ = 2d − 2. Here κ − 1 = (2d − 2)(2d − 3) since θ makes 2d − 2 turns around S1 as t goes from 0 to 2d − 2. The sequence {s1, s2, . . . , s2d−2} = θ −1(y1) is a subsequence of {ti }. Let s0 = 0 and s2d−1 = 2d − 2. Lemma 3.3 Let l ∈ {0, 1, . . . , 2d − 2}. For t ∈ (sl , sl+1) \ θ −1(D), the nets γt are equivalent. Proof Let a and b be two points in (sl , sl+1) \ θ −1(D) with a < b. If θ ([a, b]) ∩ D = ∅, then for each z ∈ CP1, ft (z) is a continuous function of t on [a, b], and, for all t ∈ [a, b], the nets γt are nondegenerate. Therefore, vi vj is a chord in γa if and only if it is also a chord in γb, and hence the two nets are equivalent. If θ ([a, b]) ∩ D = {yi }, i = 1, then θ (a) = vi and yi = vi+1 in the ordered vertex set of γa , while yi = vi and θ (a) = vi+1 in that of γb. For each z ∈ CP1 \ {yi }, ft (z) is a continuous function of t on [a, b], and, for all t ∈ [a, b] \ θ −1(yi ), the nets γt are nondegenerate. Thus, for all m, n ∈/ {i, i + 1}, vmvn is a chord in γa if and only if it is also a chord in γb. We have to consider the following two cases. Case 1: vi vi+1 is a chord in γa . In this case, every other chord vmvn in γa satisfies m, n ∈/ {i, i + 1} and hence must also be a chord in γb. Therefore, vi vi+1 must be a chord in γb. An example of this case is given in Fig. 2a–c. Case 2: vi vi+1 is not a chord in γa . Let yj be the endpoint of the chord from vi , and yk the endpoint of that from vi+1. The continuity of ft (z) on [a, b] for any z = yi implies that yj yk is not a chord in γb. Determining Γb is thus reduced to finding a net with the vertex set {yj ≺ vi ≺ vi+1 ≺ yk} such that yj and yk are not endpoints of the same chord. There are only two nets having four vertices, and exactly one of them does not have a chord joining yj to yk . For an example, see Fig. 2c–e. Remark 3.4 A similar argument can be used to determine the degenerate net Γti . If, for t close to ti , there is a chord of Γt connecting θ (t ) and θ (ti ) (see, for example, Fig. 2a, c), then the vertex θ (ti ) is of degree 2 in Γti (Fig. 2b). Otherwise, θ (ti ) is of degree 6 in Γti (Fig. 2d). Lemma 3.5 Let [γ ]l = ([Γ ]l , y1) denote the class {γt : t ∈ (sl , sl+1) \ θ −1(D)}. Then [γ ]l+1 = Shift([γ ]l ). Proof Choose representatives γt and γt of [γ ]l and [γ ]l+1, respectively, with y2d−2 ≺ θ (t ) ≺ y1 ≺ θ (t ) ≺ y2. As t goes from t to t , θ (t ) crosses the distinguished vertex y1, so Lemma 3.3 cannot be applied to γt and γt . Instead we shall consider γ = (Γt , y2) and γ = (Γt , y2) which are related to γt and γt by γt = Shift(γ ) and γt = Shift2(γ ). Since θ (t ) does not cross y2 as t goes from t to t , we can apply Lemma 3.3 to γ and γ to get that γ = γ . Thus, γt = Shift2(γ ) = Shift2(γ ) = Shift(γt ). Example 3.6 The nets a, c, e, g in Fig. 2 all belong to [γ ]0, while the net k belongs to [γ ]1 and is the shift of [γ ]0. An anonymous referee pointed out the correspondence between the (inverse of) the Shift operator and the Schutzenberger promotion p. If γ = Shift(γ ) and Y and Y are the Young tableaux corresponding to γ and γ , respectively, then p(Y ) = Y . This correspondence is described in [ 14 ] where it is attributed to White. Furthermore, this correspondence, as well as Lemma 3.3, follow from Remark 3.7 in [ 15 ]. Corollary 3.7 If t ∈ (t0, t1) and t ∈ (tκ−1, tκ ), then γt = γt . Equivalently, [γ ]0 = Proof As t goes from t to t , θ (t ) makes 2d − 2 turns around the circle. By Lemma 3.5, each complete turn of θ corresponds to the shift operator (i.e., rotation by π/(d − 1)) applied to the net. Applying the shift operator 2d − 2 times results in a complete rotation of the net, resulting in the original net. Remark 3.8 It could happen that [γ ]0 = [γ ]l for some positive integer l < 2d − 2. This occurs when [γ ]0 has rotational symmetry of order n for some integer n > 1, in which case l = (2d − 2)/n, a factor of 2d − 2. For example, the net [γ ]0 in Fig. 2a has rotational symmetry of order 3, so [γ ]0 = [γ ]2. Example 3.9 Figure 2 describes how a given net is modified as θ (t ) makes a full turn around the circle. As θ (t ) moves in (y1, y2) (Fig. 2a), the net remains unchanged. When θ (t ) reaches y2 (Fig. 2b), the net becomes degenerate with deg(y2) = 2. As θ (t ) moves away from y2 (Fig. 2c), we recover the original net until θ (t ) reaches y3 (Fig. 2d). At this point the net becomes degenerate again with deg(y3) = 6. As θ (t ) moves away from y3 (Fig. 2e), we recover the original net, and the process continues in a similar manner until θ (t ) crosses y1. The net obtained at this step (Fig. 2k) is the shift of the net in Fig. 2a. 4 Lower Bounds for k = 1 In this section, we derive lower bounds on the number of real solutions to Problem 1.5 in the special case where the interval (r, s), or its complement, contains only one fixed critical point of f . To simplify notation, we shall assume that y1 = ∞ is the only fixed vertex not contained in the interval (r, s). Throughout this section, f is a real rational function of degree d, and its net is given by Γ = f −1(RP1). Theorem 4.1 Let r < y2 < y3 < · · · < y2d−3 < s be points in R, and let y1 = ∞. Then there are at least ud − 2ud−1 classes of real rational functions f of degree d having critical points at yj , 1 ≤ j ≤ 2d − 3, and satisfying f (r) = f (s). Proof Let C be an equivalence class of the nets with 2d − 2 vertices and no interior edges connecting the distinguished vertex y1 to any of its two neighboring vertices. The number of such classes is ud − 2ud−1, since there are ud−1 classes of nets having two given neighboring vertices connected by an edge. It remains to show that for each such class C, there exist y ∈ R and a real rational function fy having the net γy ∈ C with the vertex set {y1, y2, . . . , y2d−3, y}, satisfying fy (r) = fy (s). Let y ∈ (s, ∞). There exists a unique class Fy of real rational functions with the net belonging to C and critical points y1, y2, . . . , y2d−3, y [ 4 ]. Choose fy ∈ Fy so that fy has a double pole at ∞ and fy (x) < 0 for large |x|. Let ρ : R −→ Q be given by ρ(y) = (x − y2)(x − y3) . . . (x − y2d−3)(x − y). Note that ρ(y) is the Wronskian of fy . By Lemma 2.4, ρ can be lifted to a path η : R −→ Rd with η(y) = fy satisfying properties (i)–(iii) of Lemma 2.4. By Lemma 3.3, γy ∈ C for all y ∈ The map φx : R −→ RP1 given by φx (y) = fy (x) is continuous for all x in R − {yj }j2d=−23. The continuity of φx (y) on (s, ∞) implies that for all y ∈ (s, ∞), fy (x) < 0 for large |x|. The following lemma completes the proof. Lemma 4.2 There exists y ∈ R such that fy (r) = fy (s). Proof Assume that fy (r) = fy (s) for all y ∈ [r, s]. If y ∈ [r, s], then fy cannot have a pole at r nor at s since each of [s, ∞) and (−∞, r] belongs to the boundary of a face of Γy and fy has a pole at ∞. Since both φr and φs are continuous, this implies that either −∞ < fy (r) < fy (s) < ∞ for all y ∈ [r, s] or −∞ < fy (s) < fy (r) < ∞ for all y ∈ [r, s]. Without loss of generality, we may assume that −∞ < fy (r) < fy (s) < ∞ for all y ∈ [r, s]. In particular, −∞ < fs (r) < fs (s) < ∞. For y ∈ (s, ∞), fy (y) is finite since fy has a pole at ∞ and (y, ∞) belongs to the boundary of a face of Γy . Moreover, fy (s) cannot exceed fy (y) without first decreasing to −∞ since φs is continuous and fy has a local maximum at y and no critical points between s and y. On the other hand, since Γy has no interior edges connecting ∞ to y for all y ∈ [s, ∞), limy→∞ fy (y) = −∞, and it follows that fy (s) must also decrease to −∞ on a subinterval of (s, ∞). In particular, there exists y ∈ (s, ∞) such that fy (r) = fy (s). 5 Upper and Lower Bounds for the Arc Length of f ([r, s]) In this section, we assume that f maps S1 to S1. We derive lower and upper bounds for the arc length of f ([r, s]), in terms of the net of f and the position of the points r and s relative to the vertices of the net. Consider S1 with the standard orientation. Given any two points r and s in S1, we define [r, s] and (r, s), respectively, to be the closed and open positively oriented arcs in S1 starting at r and ending at s. The standard orientation on S1 induces an order on [r, s] given by a ≤ b if and only if a ∈ [r, b]. Definition 5.1 Let Γ be a net such that all its vertices y1 < y2 < · · · < yn inside (r, s) are simple. The Young tableau of Γ corresponding to (r, s), denoted by YΓ (r, s), has 2 rows and n entries defined as follows. The integer i is placed in the first row of YΓ (r, s) if and only if yj is not connected by a chord of Γ to yi , i > j . Note that YΓ (r, s) is part of the Young tableau of (Γ , y1). See examples in Fig. 3 and Fig. 4. Definition 5.2 Given a rational function f and an oriented segment or a closed loop c in CP1 with f (c) ⊂ S1, let L(c) = Lf (c) be the argument increment of f on c. Let Γ be a net with simple vertices y1 < · · · < yn inside (r, s). Let f be a rational function with the net Γ . Assume that f is orientation preserving on (r, y1). Let Y = YΓ (r, s). Definition 5.3 Let E and O be the numbers of even and odd entries in the second row of Y . Let m be the number of vertices of Γ inside (r, s) connected by a chord to vertices outside (r, s). Lemma 5.4 The number m = n − 2(O + E) is the difference between the length of the first row of Y and the length of its second row. Consider first the case m = 0. Lemma 5.5 The following are equivalent: (i) m = 0; (ii) Y is rectangular; (iii) no vertex in (r, s) is connected by a chord of Γ to a vertex outside (r, s). Definition 5.6 Assuming that m = 0, let V = {y1, . . . , yn} be the set of vertices of Γ inside (r, s). Let C be the set of chords of Γ having both endpoints in V , and let F = F (r, s) be the set of faces F of Γ satisfying ∂F ∩ S1 ⊂ [r, s]. A chord in C will be denoted by yi yj where yi and yj are its endpoints, with i < j . Given a chord yi yj ∈ C, let Fj be the face in F having yj as its largest vertex. Let F0 be the face of Γ whose boundary contains [r, y1]. There is a one-to-one correspondence between the sets C and F . Condition (iii) of Lemma 5.5 implies that ∂F0 must also contain [yn, s]. Example 5.7 For the net Γ and arc (r, s) in Fig. 3, n = 8, E = 2, O = 2, m = 0, C = {y1y2, y3y8, y4y5, y6y7}, and F = {F2, F5, F7, F8}. Definition 5.8 Assuming that m = 0, a chord yi yj in C is even (resp., odd) if j is even (resp., odd). By the construction of YΓ (r, s) we have the following. Lemma 5.9 The numbers E and O in Definition 5.3 are, respectively, the numbers of even and odd chords in C. Definition 5.10 A face F of Γ is positive if f is orientation preserving on ∂F , otherwise F is negative. Let P and N be the numbers of positive and negative faces in F . The face F0 is always positive since f is orientation preserving on (r, y1). Since f maps the boundary of each face F ∈ F bijectively onto S1, it follows that L(∂F ) = 2π if F is positive and L(∂F ) = −2π if F is negative. Lemma 5.11 Assuming that m = 0, a chord yi yj in C is odd if and only if Fj is positive. In particular, O = P and E = N . Proof For yi yj ∈ C, the arc [yj−1, yj ] of S1 belongs to ∂Fj , and its orientation induced by Fj agrees with the standard orientation on S1. Thus, since f preserves orientation on [r, y1] and has only simple critical points in (r, s), it preserves orientation on [yj−1, yj ] if and only if j is odd. Lemma 5.12 Let m = 0. Then L([r, s]) ∈ (2π(O − E), 2π(O − E + 1)). Proof Let Ω be the region bounded by the arc [y1, yn] ⊂ S1 and α, the positively oriented curve on ∂F0 from y1 to yn (see Fig. 3). Since Ω \ Γ is the union of the faces in F , L(∂Ω) = L(∂F ) = 2π(P − N ) = 2π(O − E). On the other hand, L([r, y1]) + L(α) + L([yn, s]) ∈ (0, 2π ) since [r, y1] ∪ α ∪ [yn, s] is part of the boundary of the positive face F0 having positive orientation. Therefore, F ∈F L([r, s]) = L [r, y1] + L [y1, yn] + L [yn, s] = L [r, y1] + L(α) + L(∂Ω) + L [yn, s] ∈ 2π(O − E), 2π(O − E + 1) . Remark 5.13 Assume that m = 0. If f is orientation reversing on (r, y1), then g = 1/f is orientation preserving on (r, y1), and Lg([r, s]) = −Lf ([r, s]). Replacing f by g, we obtain from Lemma 5.12 that Lf ([r, s]) ∈ (2π(E − O − 1), 2π(E − O)). For the arc (y0, yj1 ), f is orientation preserving on (y0, y1), so by Lemma 5.12, L([y0, yj1 ]) ∈ (2π(O0 − E0), 2π(O0 − E0 + 1)). For (yj1 , yj2 ), f is orientation reversing on (yj1 , yj1+1). Since j1 is odd, the vertices that are even with respect to (yj1 , yj2 ) belong to O1, while the vertices that are odd with respect to (yj1 , yj2 ) belong to E1. Accordingly, by Remark 5.13, Lf ([yj1 , yj2 ]) ∈ (2π(O1 − E1 − 1), 2π(O1 − E1)). Applying the same arguments to (yji , yji+1 ), we get the following. Lemma 5.15 Let i ∈ {0, 1, . . . , m}. If i is even, then L([ji , ji+1]) ∈ (2π(Oi − Ei ), 2π(Oi − Ei + 1)). If i is odd, then L([ji , ji+1]) ∈ (2π(Oi − Ei − 1), 2π(Oi − Ei )). Summing up over i = 0, . . . , m, we obtain the following. Theorem 5.16 Let I = IΓ (r, s) = Then L([r, s]) ∈ I . mod 2 − n + 4O − 1 π, (n + 1) mod 2 + n − 4E + 1 π . Now we consider the case where W (p, q) has a double root y and the net Γ is degenerate. If y ∈/ (r, s), this does not affect L([r, s]). If y ∈ (r, s), then in labeling the vertices y1, . . . , yn in (r, s), we assign to y two consecutive indices and define YΓ (r, s) to be YΓ (r, s) where Γ is a nondegenerate net with the vertices and chords close to those of Γ . With this agreement, all the arguments above can be applied to Γ , and the statement of Theorem 5.16 remains true except for the case where the degree of y is 6 and both chords of Γ with the ends at y have other ends outside (r, s). This means that y represents a segment (yji , yji+1 ) of length zero. Theorem 5.17 Let y be a double root of W (p, q) inside (r, s) which is a vertex of Γ of degree 6 such that both chords of Γ with the ends at y have other ends outside (r, s). Let I = IΓ (r, s) be mod 2 − n + 4O − 1 π, (n + 1) mod 2 + n − 4E − 1 π if the number of vertices between r and y is odd, and mod 2 − n + 4O + 1 π, (n + 1) mod 2 + n − 4E + 1 π otherwise. Then L([r, s]) ∈ I . We can similarly compute IΓ (s, r) and use it to improve IΓ (r, s) in some cases. Let f be a function in Sd all of whose critical points are simple except for possibly one double critical point. Let the critical points of f be y1, . . . , y2d−2 (counted with multiplicity) with y1 ≺ r ≺ y2 y3 · · · yk+1 ≺ s ≺ yk+2 · · · y2d−2 y1. Assume that f is orientation preserving on (y1, y2). Let (Γ , y1) be the net of f with respect to y1, and let Y be the corresponding Young tableau. Let F be the set of all faces of Γ . Let E and O be the numbers of even and odd entries in the second row of Y , respectively. Then L [r, s] + L [s, r] = ∂F = 2π(E − O − 1). F ∈F So if IΓ (r, s) = (a, b), IΓ (s, r) = (a , b ) and L([r, s]) + L([s, r]) = c, then IΓ (r, s) can be improved to (a, b) ∩ (c − b , c − a ). 6 Lower Bounds in the General Case In this section, we apply the results of Sect. 5 to derive lower bounds on the number of real solutions to Problem 1.5. Without loss of generality, we may set r = −1 and s = 1. Let D = {y1, y2, . . . , y2d−3} be a set of distinct points in S1 all different from ±1. Assume that y1 does not belong to the arc [r, s]. Let f0 be a function in Sd having critical set D ∪ {1}. Let θ : R −→ S1 be the path given by θ (t ) = e2πit . By Lemma 2.4 and Notation 2.7, there exists a path η : R −→ Sd given by η(t ) = ft , where ft = pt /qt is a rational function having critical points y1, y2, . . . , y2d−3, θ (t ) counted with multiplicity, satisfying properties (i)–(iii) of Lemma 2.4. |Lft∗ ([r, s]) − limt→t∗ Lft ([r, s])| = 2π . Lemma 6.1 The map R −→ R given by t → Lft ([r, s]) is continuous everywhere except on a subset D of θ −1(D) given by D = {t ∈ θ −1(D) : pt (x) = qt (x) = 0 for some x ∈ D}. At any point of discontinuity t ∗, limt→t∗ Lft ([r, s]) exists, and Proof The first statement follows from Lemma 2.4. The second statement follows from Lemmas 2.4 and 2.5 and from the fact that ft∗ (x) is continuous at θ (t ∗) as a function of x. Let L : R −→ R be given by L(t ) = Lft ([r, s]) if t ∈/ D, limx→t Lfx ([r, s]) if t ∈ D. Corollary 6.2 The map L is continuous on R. For any t ∈ R, L(t ) ∈ 2π Z if and only if ft (r) = ft (s). For t ∈ R, let γt = (Γt , y1) be the net corresponding to ft . The net Γt is nondegenerate if and only if θ (t ) ∈/ D. If t ∈ D, then θ (t ) is a vertex of degree 2 in Γt , and if t ∈ θ −1(D) \ D, then θ (t ) is of degree 6 in Γt . This follows from Lemma 2.4 and Remark 2.9. Lemma 6.3 The map t → γt is periodic. The period T is a factor of 2d − 2. It is equal to the number of distinct nondegenerate nets in the image. Proof This follows from Corollaries 3.3 and 3.7 and Remark 3.4. 1 ) and Wn = (n + 21 , n + 1) so that θ (Vn) = (s, r) For n ∈ Z, let Vn = (n, n + 2 and θ (Wn) = (r, s). For t ∈ R, let 2π l1(t ) and 2π u1(t ) be the lower and upper endpoints of the interval IΓt (r, s) defined in Theorems 5.16 and 5.17, and let 2π l2(t ) and 2π u2(t ) be the lower and upper endpoints of the interval IΓt (s, r) defined similarly. Let 2π c = L([r, s]) + L([s, r]). Then L(t )/(2π ) ∈ (l1(t ), u1(t )) ∩ (c − u2(t ), c − l2(t )). Lemma 6.4 The functions l1(t ) and u1(t ) are constant on each set Vn. Proof The interval IΓt (r, s) depends only on the Young tableau of Γt corresponding to (r, s) which is independent of any changes outside (r, s). Lemma 6.5 The functions l1(t ) and u1(t ) are constant on each set Wn \ (θ −1(D) \ D) and assume only finitely many values on each set Wn. Proof The arc (r, s) does not contain the distinguished vertex y1, so by Corollary 3.3, the nets γt , t ∈ Wn \ θ −1(D) are equivalent. Since θ (t ) ∈ (r, s) \ D for all t ∈ Wn \ θ −1(D), Definition 5.1 implies that the Young tableaux YΓt (r, s) are identical for all t ∈ Wn \ θ −1(D). For t ∈ Wn ∩ D, the Young tableaux YΓt (r, s) are identical to those for t ∈ Wn \ θ −1(D), as defined after Theorem 5.16. This, along with the fact that Wn ∩ (θ −1(D) \ D) is finite, proves the second part of the statement. If we let γt = (Γt , y) for some fixed vertex y ∈ (r, s), we get the following. Corollary 6.6 The functions l2(t ) and u2(t ) are constant on each set Wn and assume only finitely many values on each set Vn. Let E = E(f0) be the set of equivalence classes of functions ft , t ∈ R. The map t −→ [ft ] is periodic with period T . The value of T depends on the choice of f0, but it is always a factor of 2d − 2. Therefore, instead of calculating T for each f0, we shall consider the interval [0, 2d − 2] and deal with the issue of having counted some equivalence classes more than once at the end of this section. Let U and L be two integer-valued functions defined by 1 U n + 2 U (n) = inf u1(t ), c − l2(t ) : t ∈ Vn , = inf u1(t ), c − l2(t ) : t ∈ Wn , L(n) = sup l1(t ), c − u2(t ) : t ∈ Vn , = sup l1(t ), c − u2(t ) : t ∈ Wn . The functions U and L can be easily computed since l1, u1, l2, u2 assume only finitely many values on each Vn and Wn. In addition, the continuity of L(t ) implies the existence of tn ∈ Vn and tn+1/2 ∈ Wn such that 21π L(tn) ∈ (L(n), U (n)) and 21π L(tn+1/2) ∈ (L(n + 1/2), U (n + 1/2)). Let S = {k/2 : 0 ≤ k < 4d − 4}. Let ≺ be the cyclic order on S given by 0 ≺ 1/2 ≺ 1 ≺ · · · ≺ (4d − 5)/2 ≺ 0. For i, j ∈ S, let (i, j ) = {k ∈ S|i ≺ k ≺ j }. Definition 6.7 A point i ∈ S is called a max point if (i) ∃k ∈ S such that L(i) ≥ U (k) and L(i) > L(j ), ∀j ∈ (k, i), and (ii) ∃k ∈ S such that L(i) ≥ U (k) and L(i) ≥ L(j ), ∀j ∈ (i, k). A point i ∈ S is called a min point if (i) ∃k ∈ S such that U (i) ≤ L(k) and U (i) < U (j ), ∀j ∈ (k, i), and (ii) ∃k ∈ S such that U (i) ≤ L(k) and U (i) ≤ U (j ), ∀j ∈ (i, k). Lemma 6.8 Between any two max (resp. min) points of S, there is a min (resp. max) point. Proof Let m and m be two max points in S. Assume that L(m) ≥ L(m ). There exists k ∈ (m, m ) such that L(m) ≥ L(m ) ≥ U (k). Choose k to be the first value where U attains a minimum on (m, m ) (the order here is the order induced by the cyclic order). Then k is a min point. Let m1 < m2 < · · · < ml be the max and min points of S, and let ml+1 = m1. Lemma 6.9 Let i ∈ {1, 2, . . . , l}. If mi is a max point, then U (mi+1) ≤ L(mi ). If mi is a min point, then U (mi ) ≤ L(mi+1). Proof Let mi be a max point. Assume that U (mi+1) > L(mi ). Since mi+1 is a min point, ∃k ∈ (mi , mi+1) such that U (mi+1) ≤ L(k). So L(mi ) < L(k) < U (k). Since mi is a max point, ∃k ∈ (mi , k) such that U (k ) ≤ L(mi ) < L(k). Choose k to be the first point where U attains a minimum on (mi , k). Then k is a min point between mi and mi+1, which is impossible. The proof of the second statement is similar. To simplify notation, fix i ∈ {1, 2, . . . , l} and let j = mi and k = mi+1. If j is a max point, then L(tk) < 2π U (k) ≤ 2π L(j ) < L(tj ), so the interval (L(tk), L(tj )) contains L(j ) − U (k) + 1 multiples of 2π . The continuity of L implies that each of these multiples of 2π is attained by L at some value t ∈ (tj , tk). Similarly, if j is a min point, then each of the L(k) − U (j ) + 1 multiples of 2π in the interval (L(tj ), L(tk)) is attained by L at some value t ∈ (tj , tk). Summing up over i, we get a value V (f0) which is a lower bound for the number of times L crosses a multiple of 2π over the interval [0, 2d − 2]. At each point t where this happens, the corresponding function ft satisfies ft (r) = ft (s). Algorithm 6.10 The algorithm described above to compute V (f0) depends on the net of f0 and not on f0 itself, so we may think of it as accepting a net γ and label its output V (γ ). Example 6.11 Figure 5 shows some steps of Algorithm 6.10 applied to a net γ with 6 vertices and c = 0. The only min point of γ corresponds to W2 where U = 0, and the only max point corresponds to W5 where L = 0. So V (γ ) = 2. Let f01, f02, . . . , f0ud be nonequivalent functions in Sd having critical points at 1 and at y1, y2, . . . , y2d−3. For each i ∈ {1, 2, . . . , ud }, there exists a family fti , t ∈ [0, 2d − 2], satisfying properties (i)–(iii) of Lemma 2.4. By Theorem 2.6 and Notation 2.7, the map t −→ [fti ] is unique. Let γti = (Γti , y1) be the net corresponding to fti , and let Ti be the period of the map t −→ γti . The uniqueness of the map t −→ [fti ] implies that the map t −→ γti is unique and that Ti is well defined. Let Ci = {γji : j = 0, 1, . . . , Ti − 1}. The sets Ci are well defined and partition the set C of nets with 2d − 2 vertices. This follows from the fact that the map t −→ γti is unique and its period is a factor of 2d − 2. Let k be the number of distinct sets Ci . For i ∈ {1, 2, . . . , k}, let NCi be the number of equivalence classes of functions f in Sd whose net belongs to Ci and which satisfy f (r) = f (s). For γ ∈ C, let V (γ ) be the output of Algorithm 6.10 when the input net is γ . Theorem 6.12 The number N of classes [f ] ∈ Sd having critical points at the points y1, y2, . . . , y2d−3 and satisfying f (r) = f (s) is bounded below by γ ∈C 2Vd(−γ 2) . Proof Since Ci , i = 1, 2, . . . , k, partition C, N = . . . , k}, k i=1 NCi . For each i ∈ {1, 2, Dividing both sides by Ti and taking the sum over all classes Ci , we get k i=1 NCi = k Ti −1 V (γji ) i=1 j=0 2d − 2 = γ ∈C V (γ ) 2d − 2 . The lower bounds computed by the algorithm for 4 ≤ d ≤ 14 and the corresponding values of k appear in Table 1. The first row and column give the values of d and k, respectively. Computations in www.math.tamu.edu/~sottile/research/pages/Flags/ Data/F125/We2Xe5.5.html, www.math.tamu.edu/~sottile/research/pages/Flags/Data/ F126/Ve2We7.14.html, and www.math.tamu.edu/~secant/monotone/monotoneindivproblemview.php?DB=monotoneSecant&result_id=957 show that the lower bounds for d = 4, 5, and 6 are sharp. Since t −→ γt is periodic, V (γji ) = NCi for all j ∈ {0, 1, . . . , Ti − 1}, hence, m o S 5 . g i 12 7 Combinatorial Interpretation for k = 1, 2 In this section, we give a combinatorial interpretation of the algorithm in the cases where (r, s) contains exactly one or exactly two points of D. The first case agrees with the result obtained in Sect. 4. For t ∈ Wn, (l1(t ), u1(t )) ⊆ (c − u2(t ), c − l2(t )). For t ∈ Vn, (l1(t ), u1(t )) ⊆ (c − u2(t ), c − l2(t )) except when θ (t ) coincides with a fixed vertex y and the two edges with endpoint y have their other endpoint outside (s, r ). When k = 1 and t ∈ Vn, there is only one fixed vertex outside (s, r ). So for k = 1, it is enough to consider (l1, u1). Case 1: (r, s) contains one point. Suppose that there is only one fixed vertex y in (r, s). The Young tableaux corresponding to a net with one fixed vertex in (r, s) can have one or two squares. The only one having one square gives rise to a unique (L, U )-interval, namely (−1, 1). Each of the two tableaux with two squares gives rise to two intervals. These intervals are (−1, 0), (0, 1), (−2, 0), and (0, 2). All of these 5 intervals appear in Fig. 6. The three sequences (0, 1)(−1, 1)(−1, 0), (0, 1)(−1, 1)(−2, 0), (0, 2)(−1, 1)(−1, 0), and their reversals cannot occur in the sequence of (L, U )-intervals of any net with d > 2. So (−1, 0) and (0, 1) do not yield any max or min points. The (L, U )-interval (0, 2) gives a max point if and only if it is preceded by (−2, 0)(−1, 1). In this case the net corresponding to (−1, 1) has the property that the moving vertex is outside (r, s) and the fixed vertex in (r, s) is not connected to any of its neighboring vertices. The same holds when (−2, 0) is a min point. Also, if the net has the property that the moving vertex is outside (r, s) and the fixed vertex in (r, s) is not connected to any of its neighboring vertices, then its (L, U )-interval must be preceded by (0, 2) and followed by (−2, 0) or vice versa. So given a net Γ with 2d − 2 vertices, V (Γ ) is precisely the number of vertices y of Γ not connected to any of their neighboring vertices. The lower bound for a given d and k = 1 is the number of distinct nets (Γ , y) such that Γ has 2d − 2 vertices and y is not connected to any of its neighboring vertices. This agrees with the result obtained in Sect. 4. Example 7.1 Figure 6 shows some steps of Algorithm 6.10 applied to a net γ with 6 vertices and k = 1. The degenerate nets with degenerate vertex outside (r, s) are not shown. Below each net is its Young tableau corresponding to (r, s). The last row gives the (L, U )-interval for each Vi and Wi , i = 0, 1, . . . , 5. The only min point of γ corresponds to W2 where U = 0, and the only max point corresponds to W5 where L = 0. So V (γ ) = 2, which is the number of vertices of Γ not connected to any of the neighboring vertices. Case 2: (r, s) contains two points. Suppose that there are two fixed vertices y1 and y2 in (r, s). All possible (L, U )intervals appear in Fig. 5. Any max point must correspond to an (L, U )-interval with L = 0. More specifically, any max point must correspond to an (L, U )-interval equal to (0, 2) since the interval (0, 1) must always be preceded by (0, 2). Similarly, a min point must correspond to an (L, U )-interval equal to (−2, 0). Any interval giving rise to a max (resp. min) point must be immediately followed by the interval (0, 1) (resp. (−1, 0)). The interval (0, 1) (resp. (−1, 0)) is obtained when the moving critical point is outside (r, s), the two critical points in (r, s) are connected by a chord, and the function is orientation reversing (resp. preserving) in a neighborhood of r . So the min/max points correspond to chords in the net connecting adjacent critical points and separated by an odd number of critical points. The lower bound is N2d−2, the number of nets γ = (Γ , y1) with 2d − 2 vertices satisfying y1y2 is a chord of Γ and the next chord connecting two consecutive vertices is of the form y2k y2k+1 with 1 < k < d − 1. Counting the number of such nets for a fixed k is equivalent to finding the number Nk of nets γ = (Γ , v1) having 2d − 6 vertices v1, v2, . . . , v2d−6 such that if vi vi+1 is a chord of Γ , then i ≥ 2k − 3. Let S be the set of all nondegenerate nets γ having 2d − 6 vertices. Given a net γ , let Cγ be the set of chords of γ . N2 = ud−2 since, when k = 2, there is no condition on the chords of γ . N3 is the number of nets γ such that v1v2 and v2v3 are not chords in γ . So N3 = ud−2 − 2ud−3. N4 is the number of nets γ such that vi vi+1 is not a chord in γ for i = 1, 2, 3, 4: 4 i=1 N4 = |S| − {γ ∈ S : vi vi+1 ∈ Cγ } + γ ∈ S : {v1v2, v3v4} or {v1v2, v4v5} or {v2v3, v4v5} ⊂ Cγ = ud−2 − 4ud−3 + 3ud−4. Similar computations yield N5 = ud−2 − 6ud−3 + 10ud−4 − 4ud−5, N6 = ud−2 − 8ud−3 + 21ud−4 − 20ud−5 + 5ud−6, etc. Using these values, we can compute N2d−2. 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Monique Azar, Andrei Gabrielov. Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties, Discrete & Computational Geometry, 2010, 636-659, DOI: 10.1007/s00454-010-9314-8