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53
Eigenvalues, invariant factors, highest weights, and Schubert calculus
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Gra ..."
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Cited by 176 (3 self)
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Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Contents 1. Eigenvalues of sums of Hermitian and real symmetric matrices 2. Invariant factors 3. Highest weights 4. Schubert calculus
Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro
, 2000
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Numerical Irreducible Decomposition using PHCpack
, 2003
"... Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the ..."
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Cited by 26 (15 self)
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Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.
A Family of Sparse Polynomial Systems Arising in Chemical Reaction Systems
, 1999
"... A class of sparse polynomial systems is investigated which is dened by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace i ..."
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Cited by 24 (2 self)
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A class of sparse polynomial systems is investigated which is dened by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace is of particular interest. We show that the simplest cases are equivalent to binomial systems while in general the solution structure is highly determined by the properties of the two graphs. First we recall results by Feinberg and give rigorous proofs. Secondly, we explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. The results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results.
Some Remarks on Real and Complex Output Feedback
, 1998
"... . We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geom ..."
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Cited by 23 (10 self)
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. We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real. 1. Preliminaries Let F be an arbitrary field and let m; p; n be fixed positive integers. Let A; B; C be matrices with entries in F of sizes n \Theta n, n \Theta m, and p \Theta n respectively. Identify the space of monic polynomials having degree n, s n + a n\Gamma1 s n\Gamma1 + \Delta \Delta \Delta + a 1 s + a 0 2 F[s]; with the vector space F n . In its simplest form, the static output pole placement problem asks for conditions on the matrices A; B; C which guarantee that the pole placement map Ø (A;B;C) : F mp \Gamma! F n ; K 7\Gamma! det(sI \Gamma A \Gamma BKC) (1) is surjectiv...
Numerical Evidence For A Conjecture In Real Algebraic Geometry
, 1998
"... Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are over ..."
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Cited by 23 (5 self)
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Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are overcome if we work in the true synthetic spirit of the Schubert calculus by selecting the numerically most favorable equations to represent the geometric problem. Since a wellconditioned polynomial system allows perturbations on the input data without destroying the reality of the solutions we obtain not just one instance, but a whole manifold of systems that satisfy the conjecture. Also an instance that involves totally positive matrices has been verified. The optimality of the solving procedure is a promising first step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.
GALOIS GROUPS OF SCHUBERT PROBLEMS Via Homotopy Computation
, 2009
"... Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois grou ..."
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Cited by 22 (18 self)
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Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3planes in C 8 meeting 15 fixed 5planes nontrivially is the full symmetric group S6006.
Frontiers of reality in Schubert calculus
 Bulletin of the AMS
"... Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fu ..."
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Cited by 21 (9 self)
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Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.
Robust certified numerical homotopy tracking
 In preparation
, 2010
"... Abstract. Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuatio ..."
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Cited by 17 (3 self)
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Abstract. Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with the implementation in the numerical algebraic geometry package of Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair. The numerical homotopy continuation methods are the backbone of the area of numerical algebraic geometry; while this area has a rigorous theoretic base, its existing software relies on heuristics to perform homotopy tracking. This paper has two main goals: • On one hand, we intend to overview some recent developments in the analysis of complexity of polynomial homotopy continuation methods with the view towards a practical implementation. In the last years, there has been much progress in the understanding of this problem. We hereby summarize the main results obtained, writting them in a unified and accesible way. • On the other hand, we present for the first time an implementation of a certified homotopy method which does not rely on heuristic considerations. Experiments with this algorithm are also presented, providing for the first time a tool to study deep conjectures on the complexity of homotopy methods (as Shub & Smale’s conjecture discussed below) and illustrating known –yet somehow surprising – features about these methods, as
The special Schubert calculus is real
 Electronic Research Announcements of the AMS 5:35–39
, 1999
"... Abstract. We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real. ERA of the AMS 5 (1999), pp. 35–39. Fulton asked how many solut ..."
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Cited by 17 (9 self)
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Abstract. We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real. ERA of the AMS 5 (1999), pp. 35–39. Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures [5]. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real [10]. Recently, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real [2]. Similarly, given any problem of enumerating lines in projective space incident on some general fixed linear subspaces, there are real fixed subspaces such that each of the (finitely many) incident lines are real [13]. Other examples are shown in [12, 14], and the case of 462 4planes meeting 12 general 3planes in R 7 is due to an heroic symbolic computation [4]. For any problem of enumerating pplanes having excess intersection with a collection