## Numerical Schubert Calculus (1997)

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Citations: | 44 - 24 self |

### BibTeX

@MISC{Huber97numericalschubert,

author = {Birkett Huber and Frank Sottile and Bernd Sturmfels},

title = {Numerical Schubert Calculus},

year = {1997}

}

### Years of Citing Articles

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### Abstract

### Citations

1295 |
Commutative Algebra with a View Toward Algebraic Geometry
- Eisenbud
- 1994
(Show Context)
Citation Context ...or the case (m; p) = (3; 2). The Grassmannian of 2-planes in C 5 has dimension 6 and is embedded into P 9 . Its degree (3) is five. The Grobner homotopy works directly in the ten Plucker coordinates: =-=[12]-=-; [13]; [14]; [15]; [23]; [24]; [25]; [34]; [35]; [45]: The ideal I 3;2 of the Grassmannian in the Plucker embedding is generated by five quadrics: [14][23] \Gamma [13][24] + [12][34]; [15][23] \Gamma... |

865 | Symmetric functions and Hall polynomials - Macdonald - 1979 |

219 | Young tableaux, with Applications to Representation Theory and Geometry - Fulton - 1997 |

150 | Numerical Continuation Methods: An Introduction - Allgower, Georg - 1990 |

148 | Young Tableaux - Fulton - 1997 |

141 |
The transversality of a general translate
- Kleiman
- 1974
(Show Context)
Citation Context ... transversally. In this case the following identity in the cohomology ring holds: [Y " Z] = [Y ] \Delta [Z]; where [W ] denotes the cycle class of a subvariety W . By Kleiman's Transversality The=-=orem [19], sub-=-varieties of Grass(p; m + p) in general position meet generically transversally. Transversality and generic transversality coincide when Y " Z is finite. Proposition 3.1 (Hodge and Pedoe, 1952, T... |

138 | Algorithms in Invariant Theory - Sturmfels - 1993 |

127 |
Solving polynomial systems using continuation for engineering and scientific problems
- Morgan
- 1987
(Show Context)
Citation Context ...g all d solution planes from the input data K 1 ; : : : ; K n . This amounts to solving certain systems of polynomial equations. Our algorithms are based on the paradigm of numerical homotopy methods =-=[26, 1, 2]-=-. Homotopy methods have been developed for the following classes of polynomial systems: 1. complete intersections in affine or projective spaces [11, 14], 2. complete intersections in products of proj... |

112 |
A Polyhedral method for solving sparse polynomial systems
- Huber, Sturmfels
- 1995
(Show Context)
Citation Context ...of polynomial systems: 1. complete intersections in affine or projective spaces [11, 14], 2. complete intersections in products of projective spaces [27], 3. complete intersections in toric varieties =-=[41, 17]-=-. In these cases the number of paths to be traced is optimal and equal to the standard combinatorial bounds: 1. the B'ezout number (= the product of the degrees of the equations) 2. the generalized B'... |

79 |
Homotopies exploiting Newton polytopes for solving sparse polynomial systems
- Verschelde, Verlinden, et al.
- 1994
(Show Context)
Citation Context ...of polynomial systems: 1. complete intersections in affine or projective spaces [11, 14], 2. complete intersections in products of projective spaces [27], 3. complete intersections in toric varieties =-=[41, 17]-=-. In these cases the number of paths to be traced is optimal and equal to the standard combinatorial bounds: 1. the B'ezout number (= the product of the degrees of the equations) 2. the generalized B'... |

75 | Numerical path following - Allgower, Georg - 1997 |

71 | Gröbner bases and convex polytopes, volume 8 of University Lecture Series - Sturmfels - 1996 |

65 | Determinantal rings - Bruns, Vetter - 1988 |

42 |
polyhedra and the genus of complete intersections
- Newton
- 1978
(Show Context)
Citation Context ...l and equal to the standard combinatorial bounds: 1. the B'ezout number (= the product of the degrees of the equations) 2. the generalized B'ezout number for multihomogeneous systems 3. the BKK bound =-=[4, 21, 18]-=- (= mixed volume of the Newton polytopes) None of these known homotopy methods is applicable to our problem, as the following simple example shows: Take m = 3, p = 2, and k 1 = \Delta \Delta \Delta = ... |

41 |
The number of roots of a system of equations, Funct
- Bernstein
- 1975
(Show Context)
Citation Context ...l and equal to the standard combinatorial bounds: 1. the B'ezout number (= the product of the degrees of the equations) 2. the generalized B'ezout number for multihomogeneous systems 3. the BKK bound =-=[4, 21, 18]-=- (= mixed volume of the Newton polytopes) None of these known homotopy methods is applicable to our problem, as the following simple example shows: Take m = 3, p = 2, and k 1 = \Delta \Delta \Delta = ... |

37 | Introduction to numerical algebraic geometry - Sommese, Verschelde, et al. - 2005 |

32 | Enumerative geometry for the real Grassmannian of lines in projective space - Sottile - 1997 |

31 |
Pole assignment by output feedback, in Three decades of mathematical system
- Byrnes
(Show Context)
Citation Context ...the complex plane. We remark that in practice, M may be chosen at random. 5. Applications The algorithms of Sections 2 and 3 are useful for studying both the pole assignment problem in systems theory =-=[8]-=- and real enumerative geometry [33]. We describe the connection to the control of linear systems following [8]. Suppose we have a system (for example, a mechanical linkage) with inputs u 2 R m and NUM... |

25 | A Maple package for symmetric functions
- Stembridge
- 1995
(Show Context)
Citation Context ... h kn 2 I: (2) Thus we can compute the number d by normal form reduction modulo any Grobner basis for I. More efficient methods for computing in the ring A m;p are implemented in the Maple package SF =-=[37]-=-. In the important special case k 1 = \Delta \Delta \Delta = k n = 1 there is an explicit formula for d: d = 1! 2! 3! \Delta \Delta \Delta (p \Gamma 2)! (p \Gamma 1)! \Delta (mp)! m! (m+1)! (m+ 2)! \D... |

25 | Finding All Solutions to Polynomial Systems and Other - Garcia, Zangwill - 1979 |

25 | Singular version 1.2 User Manual - Greuel, Pfister, et al. - 1998 |

24 | Some remarks on real and complex output feedback
- Rosenthal, Sottile
- 1998
(Show Context)
Citation Context ...s result, they deduced that the pole assignment problem is not in general solvable by radicals. Despite this success, only few non-trivial examples have been computed in the control theory literature =-=[30]-=-. An important question is whether a given system may be controlled by real output feedback [42, 7, 29]. That is, if all roots of '(s) are real, are there real feedback laws F satisfying (26)? Real en... |

21 | Pole assignment by output feedback, in - Byrnes - 1989 |

18 |
a System for Computation in Algebraic Geometry and Singularity Theory, available from http://www.mathematik.uni-kl.de/ftp/pub/Math/Singular
- Greuel, G, et al.
(Show Context)
Citation Context ...ariables as follows: [ff]s[fi] if and only if ff isfi i for i = 1; : : : ; p. This partially ordered set is called Young's poset. Figure 1 shows Young's poset for (m; p) = (3; 2). [12] [13] [14] [23] =-=[15]-=- [24] [25] [34] [35] [45] Figure 1. Young's poset for (m; p) = (3; 2). Fix any linear ordering on the variables in S which refines the ordering in Young's poset, and let OE denote the induced degree r... |

18 | Numerical algebraic geometry. In The mathematics of numerical analysis (Park City - Sommese, Wampler - 1995 |

17 |
Mixed volume computation by dynamic lifting applied to polynomial system solving', Discrete Comput
- Verschelde, Gatermann, et al.
- 1996
(Show Context)
Citation Context ... specific value of m and p. Smaller weights can be found using Linear Programming, as explained e.g. in the proof of [39, Proposition 1.11]. Another method would be to adapt the "dynamic" ap=-=proach in [40]-=- to our situation. This is possible since the Grobner basis in (11) is reverse lexicographic: first deform the lowest variable to zero, then deform the second lowest variable to zero, then the third l... |

17 | Mixed-volume computation by dynamic lifting applied to polynomial system solving - Verschelde, Gatermann, et al. - 1996 |

16 |
The number of equations defining a determinantal variety
- Bruns, Schwänzl
- 1990
(Show Context)
Citation Context ... \Delta minors of [X j N ]. For a purely set-theoretic (but not scheme-theoretic) representation of\Omega N a further substantial reduction in the number of equations is possible using the results of =-=[5]. An inter-=-section Y " Z of subvarieties is generically transverse if every component of Y " Z has an open subset along which Y and Z meet transversally. In this case the following identity in the coho... |

16 |
Generic eigenvalue assignment by memoryless output feedback,Systems Control Lett.,26
- Rosenthal, Schumacher, et al.
- 1995
(Show Context)
Citation Context ...Despite this success, only few non-trivial examples have been computed in the control theory literature [30]. An important question is whether a given system may be controlled by real output feedback =-=[42, 7, 29]-=-. That is, if all roots of '(s) are real, are there real feedback laws F satisfying (26)? Real enumerative geometry [33] asks a similar question: Are there real linear subspaces K 1 ; : : : ; K n in g... |

16 |
Generic properties of the pole placement problem
- Willems, Hesselink
- 1978
(Show Context)
Citation Context ...Despite this success, only few non-trivial examples have been computed in the control theory literature [30]. An important question is whether a given system may be controlled by real output feedback =-=[42, 7, 29]-=-. That is, if all roots of '(s) are real, are there real feedback laws F satisfying (26)? Real enumerative geometry [33] asks a similar question: Are there real linear subspaces K 1 ; : : : ; K n in g... |

15 |
SAGBI bases with applications to blow-up algebras
- Conca, Herzog, et al.
- 1996
(Show Context)
Citation Context ...umber of solutions equals (1.3). The first algorithm is derived from a Grobner basis for the Plucker ideal of a Grassmannian and the second from a SAGBI basis for its projective coordinate ring. (See =-=[10]-=- or [39, Ch. 11] for an introduction to SAGBI bases). Both the Grobner homotopy and SAGBI homotopy are techniques for finding linear sections of Grassmannians in their Plucker embedding. In Section 3 ... |

15 |
Eine methode zur Berechnung sämtlicher Lösungen von
- Drexler
- 1977
(Show Context)
Citation Context ...n the paradigm of numerical homotopy methods [26, 1, 2]. Homotopy methods have been developed for the following classes of polynomial systems: 1. complete intersections in affine or projective spaces =-=[11, 14]-=-, 2. complete intersections in products of projective spaces [27], 3. complete intersections in toric varieties [41, 17]. In these cases the number of paths to be traced is optimal and equal to the st... |

15 |
A Newton polyhedron and the number of solutions of a system of k equations in k unknowns
- Kouchnirenko
- 1975
(Show Context)
Citation Context ...l and equal to the standard combinatorial bounds: 1. the B'ezout number (= the product of the degrees of the equations) 2. the generalized B'ezout number for multihomogeneous systems 3. the BKK bound =-=[4, 21, 18]-=- (= mixed volume of the Newton polytopes) None of these known homotopy methods is applicable to our problem, as the following simple example shows: Take m = 3, p = 2, and k 1 = \Delta \Delta \Delta = ... |

14 | Pieri’s formula via explicit rational equivalence, Can.J.Math.49 - Sottile - 1997 |

12 |
Decomposition Bound for Bezout Numbers
- Morgan, Sommese, et al.
(Show Context)
Citation Context ...otopy with no divergent curves. Methods for such deficient systems which reduce the number of divergent curves are developed in [23, 22, 24]. When the polynomials f 1 ; : : : ; f n have special forms =-=[27, 28]-=-, then such homotopies (23) are constructed where G(X) shares this special form. When the polynomials f 1 ; : : : ; f n are sparse, polyhedral methods [41, 17] give a homotopy. The SAGBI homotopy algo... |

12 |
Enumerative geometry for real varieties, Algebraic Geometry
- Sottile
- 1995
(Show Context)
Citation Context ...applying a sequence of delicate intrinsic deformations, called Pieri homotopies, which were introduced in [35]. Pieri homotopies first arose in the study of enumerative geometry over the real numbers =-=[34, 33]-=-. For the experts we remark that it is an open problem to find Littlewood-Richardson homotopies, which would be relevant for solving polynomial equations defined by general Schubert conditions. A main... |

12 | Laksov, Schubert calculus - Kleiman, Dan - 1972 |

9 |
Nonlinear homotopies for solving deficient polynomial systems with parameters
- Li, Wang
- 1992
(Show Context)
Citation Context ...ctice, F (X) = 0 may have fewer than Q d i solutions and we desire a homotopy with no divergent curves. Methods for such deficient systems which reduce the number of divergent curves are developed in =-=[23, 22, 24]-=-. When the polynomials f 1 ; : : : ; f n have special forms [27, 28], then such homotopies (23) are constructed where G(X) shares this special form. When the polynomials f 1 ; : : : ; f n are sparse, ... |

9 |
The random product homotopy and deficient polynomial systems
- Li, Sauer, et al.
- 1987
(Show Context)
Citation Context ... 2). The Grassmannian of 2-planes in C 5 has dimension 6 and is embedded into P 9 . Its degree (3) is five. The Grobner homotopy works directly in the ten Plucker coordinates: [12]; [13]; [14]; [15]; =-=[23]-=-; [24]; [25]; [34]; [35]; [45]: The ideal I 3;2 of the Grassmannian in the Plucker embedding is generated by five quadrics: [14][23] \Gamma [13][24] + [12][34]; [15][23] \Gamma [13][25] + [12][35]; [1... |

9 |
Beziehungen zwischen den linearen Räumen auferlegbaren charakteristischen
- Schubert
(Show Context)
Citation Context ...mma 1)! \Delta (mp)! m! (m+1)! (m+ 2)! \Delta \Delta \Delta (m+p \Gamma 1)! : (3) The integer on the right hand side is the degree of the Grassmannian in its Plucker embedding. This formula is due to =-=[31]-=-; see also [16, XIV.7.8] and Section 2.3 below. The objective of this paper is to present semi-numerical algorithms for computing all d solution planes from the input data K 1 ; : : : ; K n . This amo... |

8 | bases and convex polytopes - Gröbner - 1996 |

7 |
formula via explicit rational equivalence
- Pieri's
- 1997
(Show Context)
Citation Context ... [ff]s[fi] if and only if ff isfi i for i = 1; : : : ; p. This partially ordered set is called Young's poset. Figure 1 shows Young's poset for (m; p) = (3; 2). [12] [13] [14] [23] [15] [24] [25] [34] =-=[35]-=- [45] Figure 1. Young's poset for (m; p) = (3; 2). Fix any linear ordering on the variables in S which refines the ordering in Young's poset, and let OE denote the induced degree reverse lexicographic... |

7 | Enumerative geometry for real varieties - Sottile - 1995 |

2 |
Control theory, inverse spectral problems, and real algebraic geometry
- Byrnes
- 1982
(Show Context)
Citation Context ...Despite this success, only few non-trivial examples have been computed in the control theory literature [30]. An important question is whether a given system may be controlled by real output feedback =-=[42, 7, 29]-=-. That is, if all roots of '(s) are real, are there real feedback laws F satisfying (26)? Real enumerative geometry [33] asks a similar question: Are there real linear subspaces K 1 ; : : : ; K n in g... |

2 |
Global properties of the root-locus map
- Byrnes, Stevens
- 1982
(Show Context)
Citation Context ...e conditions are independent for generic A; B; C and distinct s i , hence nsmp is necessary for there to be any feedback laws F . The critical case of n = mp is an instance of the situation in x2. In =-=[9]-=- homotopy continuation was used to solve a specific feedback problem when (m; p) = (3; 2). From this result, they deduced that the pole assignment problem is not in general solvable by radicals. Despi... |

2 |
Some conbinatorial aspects of the Schubert calculus, in Combinatoire et Représentation du Groupe Symétrique
- Stanley
- 1977
(Show Context)
Citation Context ...4]; [24]; [25]; [35]; [45]g ; f[12]; [13]; [14]; [24]; [34]; [35]; [45]g ; f[12]; [13]; [23]; [24]; [34]; [35]; [45]g ; f[12]; [13]; [23]; [24]; [25]; [35]; [45]g g A standard result in combinatorics =-=[36] stat-=-es that the cardinality of C m;p equals the number (3). From Proposition 2.1 we read off the following prime decomposition which generalizes (8): in OE (I m;p ) = " C2Cm;p h [ff] : [ff] 62 C i: (... |

1 |
Numerical Continuation Methods, An Introduction, no
- Allgower, Georg
- 1990
(Show Context)
Citation Context ...g all d solution planes from the input data K 1 ; : : : ; K n . This amounts to solving certain systems of polynomial equations. Our algorithms are based on the paradigm of numerical homotopy methods =-=[26, 1, 2]-=-. Homotopy methods have been developed for the following classes of polynomial systems: 1. complete intersections in affine or projective spaces [11, 14], 2. complete intersections in products of proj... |

1 |
Numerical path following. To appear in the Handbook of Numerical Analysis, edited by
- Allgower, Georg
- 1997
(Show Context)
Citation Context ...g all d solution planes from the input data K 1 ; : : : ; K n . This amounts to solving certain systems of polynomial equations. Our algorithms are based on the paradigm of numerical homotopy methods =-=[26, 1, 2]-=-. Homotopy methods have been developed for the following classes of polynomial systems: 1. complete intersections in affine or projective spaces [11, 14], 2. complete intersections in products of proj... |

1 |
An elimination algorithm for the computation of the zeros of a system of multivariate polynomial equations
- Auzinger, Stetter
- 1988
(Show Context)
Citation Context ...trices with random integral entries between \Gamma4 and 4. A degree reverse lexicographic Grobner basis is the input for some alternative numerical polynomial systems solvers (e.g. eigenvalue methods =-=[3]-=-). We note that the Grobner basis calculation did not terminate within one week in the case (m; p) = (6; 2). m p d(m; p) SABGI homotopy Grobner homotopy Grobner basis 3 2 5 !1 ! 0:5 ! 0:5 4 2 14 47 6 ... |

1 |
Zangwill, Finding all solutions to polynomial systems and other systems of equations
- ia, B
- 1979
(Show Context)
Citation Context ...n the paradigm of numerical homotopy methods [26, 1, 2]. Homotopy methods have been developed for the following classes of polynomial systems: 1. complete intersections in affine or projective spaces =-=[11, 14]-=-, 2. complete intersections in products of projective spaces [27], 3. complete intersections in toric varieties [41, 17]. In these cases the number of paths to be traced is optimal and equal to the st... |