Results 1  10
of
168
Solving Systems of Polynomial Equations
 American Mathematical Society, CBMS Regional Conferences Series, No 97
, 2002
"... Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, gam ..."
Abstract

Cited by 162 (10 self)
 Add to MetaCart
Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the wellknown text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner
Newton’s method with deflation for isolated singularities of polynomial systems
 Theor. Comp. Sci. 359
"... We present a modification of Newton’s method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolicnumeric: we produce a new polynomial system which has the original multiple solution as a regular root. We show that the number of deflation sta ..."
Abstract

Cited by 33 (9 self)
 Add to MetaCart
(Show Context)
We present a modification of Newton’s method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolicnumeric: we produce a new polynomial system which has the original multiple solution as a regular root. We show that the number of deflation stages is bounded by the multiplicity of the isolated root. Our implementation performs well on a large class of applications. 2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30. Key words and phrases. Newton’s method, deflation, numerical homotopy algorithms, symbolicnumeric computations. 1
Solving the likelihood equations
, 2004
"... Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include ..."
Abstract

Cited by 31 (7 self)
 Add to MetaCart
Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the JukesCantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined.
Algebraic Geometry
, 2002
"... Notes for a class taught at the University of Kaiserslautern 2002/2003 ..."
Abstract

Cited by 25 (0 self)
 Add to MetaCart
Notes for a class taught at the University of Kaiserslautern 2002/2003
Smooth and Algebraic Invariants of a Group Action: Local and Global Constructions
 THE JOURNAL OF THE SOCIETY FOR THE FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2007
"... We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational i ..."
Abstract

Cited by 22 (11 self)
 Add to MetaCart
We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan’s normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants.
An algorithm for lifting points in a tropical variety
 Collect. Math
"... Abstract. The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseuxvalued “lift ” of this point in the algebraic variety. This theorem is so fundamental because ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
(Show Context)
Abstract. The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseuxvalued “lift ” of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the “lifting algorithm ” using Singular and Gfan if the base field is Q. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K n+1,0). 1.
SumCracker: A package for manipulating symbolic sums and related objects
 J. Symb. Comput
"... We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many of which there has not been any software available up to now. The purpose of this paper is to illustr ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many of which there has not been any software available up to now. The purpose of this paper is to illustrate how to solve problems using that package.
Rational invariants of a group action. Construction and rewriting
, 2007
"... Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a crosssection is added to the ideal of the graph. Zerodimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.
A combinatorial approach to involution and δregularity I: Involutive Bases in . . .
, 2002
"... Involutive bases are a special form of nonreduced Gröbner bases with additional combinatorial properties. Their origin lies in the JanetRiquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also noncommutativ ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
Involutive bases are a special form of nonreduced Gröbner bases with additional combinatorial properties. Their origin lies in the JanetRiquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also noncommutative algebras like those generated by linear differential and difference operators or universal enveloping algebras of (finitedimensional) Lie algebras. A number of basic properties are derived and we provide concrete algorithms for their construction. Furthermore, we develop a theory for involutive bases with respect to semigroup orders (as they appear in local computations) and over coefficient rings, respectively. In both cases it turns out that generally only weak involutive bases exist.