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23
PHoM  a Polyhedral Homotopy Continuation Method for Polynomial Systems
 Computing
, 2003
"... PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral ..."
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Cited by 39 (10 self)
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PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f (x) = 0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f (x) = 0. The third module Verify checks whether all isolated solutions of f (x) = 0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.
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.
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.
Toric Intersection Theory for Affine Root Counting
 Journal of Pure and Applied Algebra
, 1997
"... Given any polynomial system with xed monomial term structure, we give explicit formulae for the generic number of roots (over any algebraically closed eld) with specied coordinate vanishing restrictions. For the case of ane space minus an arbitrary union of coordinate hyperplanes, these formulae ..."
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Cited by 23 (7 self)
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Given any polynomial system with xed monomial term structure, we give explicit formulae for the generic number of roots (over any algebraically closed eld) with specied coordinate vanishing restrictions. For the case of ane space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coecients must be chosen generically for our formulae to be exact. Our underlying framework provides a new toric variety setting for computational intersection theory in ane space minus an arbitrary union of coordinate hyperplanes. We thus show that, at least for root counting, it is better to work in a naturally associated toric compactication instead of always resorting to products of projective spaces. 1.
A Method for Tracking Singular Paths with Application to the Numerical Irreducible Decomposition
, 2002
"... In the numerical treatment of solution sets of polynomial systems, methods for sampling and tracking a path on a solution component are fundamental. For example, in the numerical irreducible decomposition of a solution set for a polynomial system, one first obtains a "witness point set" co ..."
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Cited by 19 (13 self)
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In the numerical treatment of solution sets of polynomial systems, methods for sampling and tracking a path on a solution component are fundamental. For example, in the numerical irreducible decomposition of a solution set for a polynomial system, one first obtains a "witness point set" containing generic points on all the irreducible components and then these points are grouped via numerical exploration of the components by path tracking from these points. A numerical difficulty arises when a component has multiplicity greater than one, because then all points on the component are singular. This paper overcomes this di#culty using an embedding of the polynomial system in a family of systems such that in the neighborhood of the original system each point on a higher multiplicity solution component is approached by a cluster of nonsingular points. In the case of the numerical irreducible decomposition, this embedding can be the same embedding that one uses to generate the witness point set. In handling the case of higher multiplicities, this paper, in concert with the methods we previously proposed to decompose reduced solution components, provides a complete algorithm for the numerical irreducible decomposition. The method is applicable to tracking singular paths in other contexts as well.
Computing All Nonsingular Solutions of Cyclicn Polynomial Using Polyhedral Homotopy Continuation Methods
 J. COMPUT. APPL. MATH
, 2001
"... All isolated solutions of the cyclicn polynomial equations are not known for larger dimensions than 11. We exploit two types of symmetric structures in the cyclicn polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method a ..."
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Cited by 19 (5 self)
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All isolated solutions of the cyclicn polynomial equations are not known for larger dimensions than 11. We exploit two types of symmetric structures in the cyclicn polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solutions. Numerical results on the cyclic8 to the cyclic12 polynomial equations, including their solution information, are given.
Isosingular Sets and Deflation
, 2011
"... This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regulari ..."
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Cited by 16 (7 self)
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This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which are similar to the sequences arising in ThomBoardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic sets that can be investigated with methods from numerical algebraic geometry.
POLYNOMIAL HOMOTOPIES FOR DENSE, SPARSE AND DETERMINANTAL SYSTEMS
, 1999
"... 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 ..."
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Cited by 12 (1 self)
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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
Toric Newton Method for Polynomial Homotopies
, 1999
"... This paper defines a generalization of Newton's method to deal with solution paths defined by polynomial homotopies that lead to extremal values. Embedding the solutions in a toric variety leads to explicit scaling relations between coefficients and solutions. Toric Newton is a symbolicnumeric ..."
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Cited by 11 (5 self)
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This paper defines a generalization of Newton's method to deal with solution paths defined by polynomial homotopies that lead to extremal values. Embedding the solutions in a toric variety leads to explicit scaling relations between coefficients and solutions. Toric Newton is a symbolicnumeric algorithm where the symbolic preprocessing exploits the polyhedral structures. The numerical stage uses the additional variables introduced by the homogenization to scale the components of the solution vectors to the complex unit circle. Toric Newton generates appropriate affine charts and enables to approximate the magnitude of large solutions of polynomial systems.
PHoMpara  parallel implementation of the Polyhedral Homotopy continuation Method for polynomial systems
, 2006
"... The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral hom ..."
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Cited by 10 (1 self)
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The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral homotopy functions, tracing the solution curves of the homotopy equations, and verifying the obtained solutions. A software package PHoMpara parallelizes PHoM to solve a polynomial system of large size. Many characteristics of the polyhedral homotopy continuation method make parallel implementation efficient and provide excellent scalability. Numerical results include some large polynomial systems that had not been solved.