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122
SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS
, 2002
"... These are the lecture notes for ten lectures to be given at the CBMS ..."
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Cited by 150 (12 self)
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These are the lecture notes for ten lectures to be given at the CBMS
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 ..."
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Cited by 29 (10 self)
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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
B.: Solving the likelihood equations
 Found. Comput. Math
"... 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 ..."
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Cited by 28 (5 self)
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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. 1
Algebraic Geometry
, 2002
"... Notes for a class taught at the University of Kaiserslautern 2002/2003 ..."
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Cited by 23 (0 self)
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Notes for a class taught at the University of Kaiserslautern 2002/2003
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 ..."
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Cited by 20 (3 self)
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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 ..."
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Cited by 19 (6 self)
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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.
Involutive Algorithms for Computing Gröbner Bases
, 2005
"... In this paper we describe an efficient involutive algorithm for constructing ..."
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Cited by 13 (6 self)
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In this paper we describe an efficient involutive algorithm for constructing
A combinatorial approach to involution and δregularity II: Structure analysis of polynomial modules with Pommaret bases
, 2002
"... Abstract 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 nonco ..."
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Cited by 12 (3 self)
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Abstract 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. 1