## Enumerative real algebraic geometry (2003)

Venue: | Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 60, AMS |

Citations: | 9 - 2 self |

### BibTeX

@INPROCEEDINGS{Sottile03enumerativereal,

author = {Frank Sottile},

title = {Enumerative real algebraic geometry},

booktitle = {Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 60, AMS},

year = {2003},

pages = {139--180}

}

### OpenURL

### Abstract

### Citations

387 |
A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys
- Candelas, Ossa, et al.
- 1991
(Show Context)
Citation Context ... [28], 609,250 conics [38], and 371,206,375 twisted cubics. The number of twisted cubics and higher degree rational curves was computed in the seminal paper of Candelas, de la Ossa, Green, and Parkes =-=[6]-=-. How many of the curves can be real in problems of this type? For example, how many real lines can there be on a real quintic hypersurface in P 4 ? A real homogeneous polynomial f(x) is positive semi... |

197 |
Mirror symmetry and algebraic geometry
- Cox, Katz
- 1999
(Show Context)
Citation Context ... to the solution of a large class of similar enumerative problems involving, among other things, the number of rational curves of asxed degree on a Calabi-Yau threefold. (See the book of Cox and Katz =-=[10]-=-.) For example, on a general quintic hypersurface in P 4 there are 2875 lines [28], 609,250 conics [38], and 371,206,375 twisted cubics. The number of twisted cubics and higher degree rational curves ... |

189 | Algebraic Complexity Theory - Bürgisser, Clausen, et al. - 1997 |

171 |
Desingularisation des varietes de Schubert generalises, Ann
- Demazure
- 1974
(Show Context)
Citation Context ... `(w 2 ) + + `(w s ) = dim F` a and generalsags F 1 ; F 2 ; : : : ; F s , what is the number of points in the intersection Xw1 F 1 \ Xw2 F 2 \ \ Xws F s ? There are algorithms [2, 13] for computing this number and the numbers for the Lagrangian Schubert calculus in Section 4.2.4. When w = (i; i + 1), XwF is the simple Schubert variety, written X i F , X i F = X (i;i+1) F =... |

168 |
Notes on stable maps and quantum cohomology, in: Algebraic geometry
- Fulton, Pandharipande
- 1995
(Show Context)
Citation Context ...plane cubics containing 8 points generalizes to the problem of enumerating rational plane curves of degree d containing 3d 1 points. Let N d be the number of such curves, which satises the recursion [22] N d = X d 1 +d 2 =d N d 1 N d 2 d 2 1 d 2 2 3d 4 3d 1 2 d 3 1 d 2 3d 4 3d 1 1 : The values N 1 = N 2 = 1 are trivially fully real, and we have just seen that N 3 = 12 is fully real. The ne... |

143 |
The transversality of a general translate
- Kleiman
- 1974
(Show Context)
Citation Context ... \sL 2 ) \ \sL s ) ; (4.3) y This indexing of the Catalan numbers is shifted from that of some other authors. 20 FRANK SOTTILE when the intersection is transverse. (A general theorem of Kleiman [42=-=]-=- guarantees transversality when the L i are in general position, and also implies transversality for the other intersections considered in this section.) There are algorithms due to Schubert [66] (whe... |

111 |
A polyhedral method for solving sparse polynomial systems
- Huber, Sturmfels
- 1995
(Show Context)
Citation Context ...em again has d complex solutions, but there are systems with any number of real solutions, so (Q d ) = d. 2.1. Polyhedral homotopy algorithm. The polyhedral homotopy algorithm of Huber and Sturmfels [=-=33-=-] deforms the sparse system (2.1) into a system where the number of solutions is evident. It gives an eective demonstration of the BKK bound and is based upon Sturmfels's generalization [82] of Viro's... |

67 | On quantum cohomology rings of Fano manifolds and a formula of Vafa and
- Siebert, Tian
- 1997
(Show Context)
Citation Context ... 2 ; : : : ; t s 2 P 1 are general and F 1 ; F 2 ; : : : ; F s are generalsags? Algorithms to compute this number were proposed by Vafa [83] and Intriligator [36] and proven by Siebert and Tian [67=-=-=-] and by Bertram [3]. A simple quantum Schubert condition denes a subvariety of codimension 1, M(t) \ L 6= f0g ; (4.16) where L is a (n k)-plane. Let d(q; k; n) be the number of maps M 2 M q k;n satis... |

66 |
Über die Darstellung definiter Formen als Formenquadraten
- Hilbert
(Show Context)
Citation Context ... type? For example, how many real lines can there be on a real quintic hypersurface in P 4 ? A real homogeneous polynomial f(x) is positive semi-definite (psd) if f(x) ≥ 0 whenever x is real. Hilbert =-=[29]-=- proved that a psd ternary quartic is a sum of three squares of real quadratic forms. In fact, a general quartic is a sum of three squares of complex quadratic forms in 63 different ways [86]. Powers ... |

62 | Quantum Schubert calculus
- Bertram
- 1997
(Show Context)
Citation Context ... 1 are general and F 1 ; F 2 ; : : : ; F s are generalsags? Algorithms to compute this number were proposed by Vafa [83] and Intriligator [36] and proven by Siebert and Tian [67] and by Bertram [3]. A simple quantum Schubert condition denes a subvariety of codimension 1, M(t) \ L 6= f0g ; (4.16) where L is a (n k)-plane. Let d(q; k; n) be the number of maps M 2 M q k;n satisfying dimM q k;n {m... |

57 | Rational functions with real critical points and the
- Eremenko, Gabrielov
(Show Context)
Citation Context ... fully in [78]. 5.1. Rational functions with real critical points. By far the strongest evidence for Conjecture 5.1 is that it is true when k or n k is equal to 2. Theorem 5.6 (Eremenko and Gabrielov =-=[16]-=-). Conjecture 5.1 is true when one of k or n k is 2. This is a consequence of a theorem about rational functions with real critical points. A rational function is an algebraic map ' : P 1 ! P 1 . Two ... |

55 | Divisors on general curves and cuspidal rational curves - Eisenbud, Harris - 1983 |

45 |
Galois groups of enumerative problems
- Harris
- 1979
(Show Context)
Citation Context ... other things, the number of rational curves of asxed degree on a Calabi-Yau threefold. (See the book of Cox and Katz [10].) For example, on a general quintic hypersurface in P 4 there are 2875 lines =-=[28]-=-, 609,250 conics [38], and 371,206,375 twisted cubics. The number of twisted cubics and higher degree rational curves was computed in the seminal paper of Candelas, de la Ossa, Green, and Parkes [6]. ... |

45 | Numerical Schubert calculus
- Huber, Sottile, et al.
- 1998
(Show Context)
Citation Context ...ection 2.2. Just as the arguments of Section 2.2 were linked to the homotopy algorithms of Huber and Sturmfels, the proof of Theorem 4.4 leads to numerical homotopy methods for solving these problems =-=[32, 34]-=-. We develop further geometric properties of Grassmann varieties. The kth exterior power of the embedding K ! C n of a k-plane K into C n gives the embedding C ' ^ k K ! ^ k C n ; (4.4) whose image is... |

43 |
polyhedra, and the genus of complete intersections
- Khovanckii, Newton
- 1978
(Show Context)
Citation Context ...ric given its structure, then it has exactly this number of solutions in (C ) n . 4 FRANK SOTTILE This result was developed in a series of papers by Kouchnirenko [45], Bernstein [1], and Khovanskii [=-=40]-=-. For simplicity of exposition, we will largely restrict ourselves to the case when the polynomials all have the same Newton polytope P . Given a polytope P with vertices in the integral lattice, what... |

41 |
The number of roots of a system of equations, Funct
- Bernstein
- 1975
(Show Context)
Citation Context ...f the system is generic given its structure, then it has exactly this number of solutions in (C ) n . 4 FRANK SOTTILE This result was developed in a series of papers by Kouchnirenko [45], Bernstein [=-=1]-=-, and Khovanskii [40]. For simplicity of exposition, we will largely restrict ourselves to the case when the polynomials all have the same Newton polytope P . Given a polytope P with vertices in the i... |

41 |
The Stewart Platform of General Geometry has 40
- Raghavan
- 1993
(Show Context)
Citation Context ...ow many (complex) positions are there for a generic choice of the distances l 1 ; l 2 ; : : : ; l 6 ? How many of these can be real? In the early 1990's, several approaches (numerical experimentation =-=[59-=-], intersection theory [62], Grobner bases [48], resultants [53], and algebra [54]) each showed that there 14 FRANK SOTTILE are 40 complex positions of a general Stewart platform. The obviously practi... |

40 |
On the finiteness of rational curves on quintic threefolds
- Katz
- 1986
(Show Context)
Citation Context ...mber of rational curves of asxed degree on a Calabi-Yau threefold. (See the book of Cox and Katz [10].) For example, on a general quintic hypersurface in P 4 there are 2875 lines [28], 609,250 conics =-=[38]-=-, and 371,206,375 twisted cubics. The number of twisted cubics and higher degree rational curves was computed in the seminal paper of Candelas, de la Ossa, Green, and Parkes [6]. How many of the curve... |

36 | Topological properties of real algebraic varieties: du côté de chez Rokhlin - Degtyarev, Itenberg, et al. |

35 |
Common tangents to four unit balls
- Macdonald, Pach, et al.
(Show Context)
Citation Context ...pite its simplicity, this question does not seem to have been asked classically, but rather arose in discrete and computational geometry y . The case n = 3 was solved by Macdonald, Pach, and Theobald =-=[5-=-1] and the general case more recently x [79]. Theorem 3.10. 2n 2 general spheres in R n (n 3) have 3 2 n 1 complex common tangent lines, and there are 2n 2 such spheres with all common tangent lines... |

31 |
Mémoire sur la théorie de l'octaèdre articulé
- Bricard
(Show Context)
Citation Context ...f the platform in space? It had long been understood that several positions were possible for a given sextuple of lengths. An early work in 1897 showed there could be as many as 16 dierent positions [=-=4]-=-. This leads to the following enumerative problem. Question 3.4. For a given Stewart platform, how many (complex) positions are there for a generic choice of the distances l 1 ; l 2 ; : : : ; l 6 ? Ho... |

30 | The Stewart-Gough platform of general geometry can have 40 real postures
- Dietmaier
- 1998
(Show Context)
Citation Context ...ny positions could be real remained open until 1998, when Dietmaier introduced a novel method tosnd a value of the distances l 1 ; l 2 ; : : : ; l 6 with all 40 positions real. Theorem 3.5 (Dietmaier =-=[14]-=-). All 40 positions can be real. Dietmaier's method willsnd future applications to other problems of this kind. He began with a formulation of the problem as a system of equations depending upon the d... |

26 |
Catalan numbers and branched coverings by the Riemann sphere
- Goldberg
- 1991
(Show Context)
Citation Context ...' has a critical point at t 2 P 1 if d' vanishes at t. If we consider the composition (5.7), this implies that the center E meets the line tangent to the rational normal curves(P 1 ) ats(t). Goldberg =-=[24]-=- asked (and answered) the question: how many equivalence classes of rational functions of degree d have a given set of 2d 2 critical points? Reasoning as above, she reduced this to the problem of dete... |

26 |
A plataform with 6 degrees of freedom
- Stewart
- 1965
(Show Context)
Citation Context ...e stated over the real numbers. 3.2. The Stewart-Gough platform. The position of a rigid body in R 3 has 6 degrees of freedom. This is exploited in robotics, giving rise to the Stewart-Gough platforms=-=[25, 80-=-]. Specically, suppose we havesxed points A 1 ; A 2 ; : : : ; A 6 in space and 6 points B 1 ; B 2 ; : : : ; B 6 on a rigid body B (the framework of Figure 5). The body is A 1 A 2 A 3 A 4 A 5 A 6 B 1 B... |

25 |
Topological Mirrors And Quantum Rings
- Vafa
- 1992
(Show Context)
Citation Context ... i ) 2 X i F i for i = 1; 2; : : : ; s ; where t 1 ; t 2 ; : : : ; t s 2 P 1 are general and F 1 ; F 2 ; : : : ; F s are generalsags? Algorithms to compute this number were proposed by Vafa [83] and Intriligator [36] and proven by Siebert and Tian [67] and by Bertram [3]. A simple quantum Schubert condition denes a subvariety of codimension 1, M(t) \ L 6= f0g ; (4.16) where L is a (n k)-pla... |

24 |
Schubert calculus: Polynomial systems and a conjecture of Shapiro and
- Real
(Show Context)
Citation Context ...[57] and Giambelli [23] to compute these numbers. Other than the case when the i are indices of special Schubert varieties, it remains open whether the general Schubert calculus is fully real. (See [=-=74-=-] and [71] for some cases.) 4.2.2. Quantum Schubert calculus. The space M q k;n of degree q maps M : P 1 ! Gr(k; n) is a smooth quasi-projective variety [9]. A point t 2 P 1 and a Schubert variety X ... |

23 |
Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control
- Huber, Verschelde
(Show Context)
Citation Context ...ection 2.2. Just as the arguments of Section 2.2 were linked to the homotopy algorithms of Huber and Sturmfels, the proof of Theorem 4.4 leads to numerical homotopy methods for solving these problems =-=[32, 34]-=-. We develop further geometric properties of Grassmann varieties. The kth exterior power of the embedding K ! C n of a k-plane K into C n gives the embedding C ' ^ k K ! ^ k C n ; (4.4) whose image is... |

22 |
Contribution to discussion of papers on research in automotive stability, control and tyre performance
- Gough
(Show Context)
Citation Context ...e stated over the real numbers. 3.2. The Stewart-Gough platform. The position of a rigid body in R 3 has 6 degrees of freedom. This is exploited in robotics, giving rise to the Stewart-Gough platforms=-=[25, 80-=-]. Specically, suppose we havesxed points A 1 ; A 2 ; : : : ; A 6 in space and 6 points B 1 ; B 2 ; : : : ; B 6 on a rigid body B (the framework of Figure 5). The body is A 1 A 2 A 3 A 4 A 5 A 6 B 1 B... |

22 |
On the number of real roots of a sparse polynomial system
- Sturmfels
- 1991
(Show Context)
Citation Context ...; (2.5) with each a i 6= 0. Here, d 1 j d 2 j j d n are the invariant factors of the matrix whose ith column is v i v 0 and d 1 d 2 d n is the normalized volume of P . Following Sturmfels [81], let e(P ) be the number of these invariant factors which are even. If e(P ) = 0, so that d is odd, then P is an odd cell. Proposition 2.5. The polynomial system (2:5) has 2 e(P ) real solutions if a... |

22 |
algebraic plane curves: constructions with controlled topology
- Viro, Real
- 1990
(Show Context)
Citation Context ...f solutions is evident. It gives an eective demonstration of the BKK bound and is based upon Sturmfels's generalization [82] of Viro's method for constructing real varieties with controlled topology [=-=85]-=-. Example 2.4. Suppose P is a n-simplex which meets the integral lattice only at its vertices. Translating one vertex to the origin, the others are linearly independent. (Translating corresponds to di... |

21 |
Schubert cells and cohomology
- Bernstein, Gelfand, et al.
- 1973
(Show Context)
Citation Context ... `(w 2 ) + + `(w s ) = dim F` a and generalsags F 1 ; F 2 ; : : : ; F s , what is the number of points in the intersection Xw1 F 1 \ Xw2 F 2 \ \ Xws F s ? There are algorithms [2, 13] for computing this number and the numbers for the Lagrangian Schubert calculus in Section 4.2.4. When w = (i; i + 1), XwF is the simple Schubert variety, written X i F , X i F = X (i;i+1) F =... |

21 | The 40 generic positions of a parallel robot
- Mourrain
- 1993
(Show Context)
Citation Context ...e distances l 1 ; l 2 ; : : : ; l 6 ? How many of these can be real? In the early 1990's, several approaches (numerical experimentation [59], intersection theory [62], Grobner bases [48], resultants [=-=53]-=-, and algebra [54]) each showed that there 14 FRANK SOTTILE are 40 complex positions of a general Stewart platform. The obviously practical question of how many positions could be real remained open u... |

18 | and X.Wang, Degree of the generalized Plücker embedding of a quot scheme and quantum
- Ravi, Rosenthal
- 1998
(Show Context)
Citation Context ...; k; n) be the number of maps M 2 M q k;n satisfying dimM q k;n {many general simple quantum Schubert conditions (4.16). A combinatorial formula for this number was given by Ravi, Rosenthal, and Wang =-=[60]-=-. For a survey of this particular enumerative problem and its importance to linear systems theory, see [77]. y In this surveysags are general when the corresponding intersection is transverse. ENUMERA... |

15 |
A Newton polyhedron and the number of solutions of a system of k equations in k unknowns
- Kouchnirenko
- 1975
(Show Context)
Citation Context ... in (C ) n . If the system is generic given its structure, then it has exactly this number of solutions in (C ) n . 4 FRANK SOTTILE This result was developed in a series of papers by Kouchnirenko [4=-=5]-=-, Bernstein [1], and Khovanskii [40]. For simplicity of exposition, we will largely restrict ourselves to the case when the polynomials all have the same Newton polytope P . Given a polytope P with ve... |

13 |
Notes towards a constructive proof of Hilbert’s theorem on ternary quartics
- Powers, Reznick
- 2000
(Show Context)
Citation Context ...t a psd ternary quartic is a sum of three squares of real quadratic forms. In fact, a general quartic is a sum of three squares of complex quadratic forms in 63 dierent ways [86]. Powers and Reznick [=-=58-=-] studied the question of how many ways one may represent a ternary quartic as a real sum or dierence of three squares. In every instance, they found that 15 of the 63 ways involved real quadratic for... |

12 |
Eine neue Relation zwischen den Singularitaten einer algebraischen Curve
- Klein
(Show Context)
Citation Context ...c with 8 realsexes is provided by the Hilbert quartic [30], which is dened by (x 2 + 2y 2 z 2 )(2x 2 + y 2 z 2 ) + z 4 =100 = 0 : We display this curve in Figure 3, marking thesexes with dots. Klein [=-=44]-=- later showed that a general real plane curve has at most 1/3 of itssexes real. Harnack [27] proved that a smooth real algebraic curve of genus g has at most g + 1 topological components, and he const... |

12 | Enumerative geometry for real varieties, Algebraic Geometry - Sottile - 1995 |

12 |
Almindelige Egenskaber ved Systemer af plane Kurver
- Zeuthen
(Show Context)
Citation Context ... 2 2 3d 4 3d 1 2 d 3 1 d 2 3d 4 3d 1 1 : The values N 1 = N 2 = 1 are trivially fully real, and we have just seen that N 3 = 12 is fully real. The next case of N 4 = 620 (computed by Zeuthen [88]) seems quite challenging. Remark 3.8. The most interesting feature of Theorem 3.6 is the existence of a lower bound on the number of real solutions, which is a new phenomenon. In Section 6 we shall s... |

12 | Maximally inflected real rational curves
- Kharlamov, Sottile
(Show Context)
Citation Context ...maximally inflected curves. See Section 5.1 for the connection. Such curves have at most g − d + 2 of their nodes real, and there exist curves with the extreme values of 0 and of g − d + 2 real nodes =-=[39]-=-. For example, a rational quartic (d = 4) has 6 flexes and g = 3 nodes. If all 6 flexes are real, then at most one node is real. Figure 4 shows maximally inflected quartics with and 0 and 1 nodes. The... |

11 |
The consistent selection of local coordinates in linear system identification
- Clark
- 1976
(Show Context)
Citation Context ...neral Schubert calculus is fully real. (See [74] and [71] for some cases.) 4.2.2. Quantum Schubert calculus. The space M q k;n of degree q maps M : P 1 ! Gr(k; n) is a smooth quasi-projective variety [9]. A point t 2 P 1 and a Schubert variety X F together impose a quantum Schubert condition on maps M 2 M q k;n , M(t) 2 X F : The set of such maps has codimension jj in M q k;n . The quantum S... |

11 |
The number of conics tangent to 5 given conics: the real case. Universite de Geneve, preprint
- Ronga, Tognoli, et al.
- 1995
(Show Context)
Citation Context ...For example, how many of the 3264 conics tangent tosve general conics can be real?" He answered this question in the armative in 1986, but did not publish that result. Later, Ronga, Tognoli, and =-=Vust [61]-=- gave a careful argument that all 3264 can be real. This example is very striking, both for the number, 3264, and because this problem of conics has long been an important testing ground for ideas in ... |

11 | Polynomial homotopies for dense, sparse and determinantal systems
- Verschelde
- 1999
(Show Context)
Citation Context ...ns tosnd soutions for the new set of distances, and then repeats this procedure until the goal is acheived (eg. the conjugate pair collides). This is an application of numerical homotopy continuation =-=[84]-=-. While there is no guarantee that this method will even successfully collide two conjugate solutions, Dietmaier uses it tosnd a sextuple of distances with all 40 solutions real. While at each step th... |

11 | Sign-balanced posets
- White
- 2001
(Show Context)
Citation Context ...k; n) (=degree of the Grassmannian) real points in 1 W (f) (one for each chain in the Bruhat order) and show that det dW = ( 1) !(q) ; at the point in 1 W (f) corresponding to the chain q. White [87=-=]-=- studied the statistic jI(k; n)( 1)j and showed that it equals zero if and only if n is even, and that jI(2; 2n)( 1)j = C n = u 2n . Eremenko and Gabrielov deduced Corollary 6.6 (Eremenko and Gabrielo... |

10 | Intersection Theory. No. 2 in Ergebnisse der Mathematik und ihrer Grenzgebiete - Fulton - 1984 |

10 | rational curves in Grassmannians - Real |

9 |
Multivariate Descartes' rule. Beitrage zur Algebra und Geometrie
- Itenberg, Roy
- 1996
(Show Context)
Citation Context ...2.2].) X F a facet of Pw 2 e(F ) : (2.6) More sophisticated accounting of the possible signs of the coecients of facial subsystems improves this bound. This accounting is accomplished in [56] and [3=-=7]-=-, leading to a combinatorial upper bound for such limiting systems. Itenberg and Roy [37] show there is a system (2.1) for which this upper bound is attained, and thus obtain Combinatorial upper bound... |

9 |
Generalized Stewart platform: how to compute with rigid motions
- Lazard
- 1993
(Show Context)
Citation Context ...eric choice of the distances l 1 ; l 2 ; : : : ; l 6 ? How many of these can be real? In the early 1990's, several approaches (numerical experimentation [59], intersection theory [62], Grobner bases [=-=48]-=-, resultants [53], and algebra [54]) each showed that there 14 FRANK SOTTILE are 40 complex positions of a general Stewart platform. The obviously practical question of how many positions could be rea... |

8 |
Numero delle involuzioni razionali gaicenti sopra una curva di dato genere
- Castelnuovo
(Show Context)
Citation Context ... consequence of Theorem 5.3, and Conjecture 5.1 predicts this is a rich class of real curves in P m . This connection between linear series and the Schubert calculus originated in work of Castelnuovo =-=[7]-=-. 5.2. Generalizations of Conjecture 5.1. The Grassmannian,sag manifolds, and Lagrangian Grassmannian are examples ofsag varieties G=P where G is a reductive algebraic group and P a parabolic subgroup... |

8 |
Sul problema degli spazi secanti
- Pieri
(Show Context)
Citation Context ... when the L i are in general position, and also implies transversality for the other intersections considered in this section.) There are algorithms due to Schubert [66] (when each l i = 1) and Pieri [57] to compute the expected number of solutions. When each l i = 1, Schubert [64] showed that the number of solutions is equal to d(n; k) := 1! 2! (k 1)! [k (n k)]! (n k)! (n k+1)! (n 1)! :... |

7 | A simple counterexample to Kouchnirenko’s conjecture
- Haas
(Show Context)
Citation Context ... i monomials has at most (m 1 1)(m 2 1) (m n 1) solutions with positive coordinates. For a system of two trinomials in 2 variables, Kouchnirenko's conjecture asserts that + 4. In 2000, Haas [26] gave the counterexample x 108 + 1:1y 54 1:1y = 0 y 108 + 1:1x 54 1:1x = 0; a system of two trinomials with 5 solutions having positive coordinates. Although Kouchnirenko 's conjecture is false, the q... |