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

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.

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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 non-trivial 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...

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 well-conditioned 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.

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 4-planes meeting 12 general 3-planes in R 7 is due to an heroic symbolic computation [4]. For any problem of enumerating p-planes having excess intersection with a collection

Abstract. 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. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real. Similarly, given any problem of enumerating p-planes incident on some general fixed subspaces, there are real fixed subspaces such that each of the (finitely many) incident p-planes are real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions be real.

. Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and applications are discussed. AMS Subject Classification. 14N10, 14M15, 52A39, 52B20, 52B55, 65H10, 68Q40. Keywords. polynomial system, numerical algebraic geometry, homotopy, continuation, deformation, path following, dense, sparse, determinantal, B'ezout bound, Newton polytope, mixed volume, root count, enumerative geometry, numerical Schubert calculus. Contents 1. Introduction 1 2. Three Classes of Polynomial Systems 3 3. The Principles of Polynomial Homotopy Continuation Methods 5 4. The Geometry of the Deformations 8 5. Root Counts and Start Systems 10 5.1. Dense Polynomials modeled by Highest Degrees 10 5.2. Mixed Subdivisions of Newton Polytopes to compute Mixed Vo...

Abstract. We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these other enumerative problems.

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.