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25
Solving Systems of Polynomial Equations
 AMERICAN MATHEMATICAL SOCIETY, CBMS REGIONAL CONFERENCES SERIES, NO 97
, 2002
"... 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, ..."
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Cited by 223 (13 self)
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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
A finiteness theorem for markov bases of hierarchical models
 J. COMB. THEORY SER. A
, 2007
"... We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We giv ..."
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Cited by 39 (4 self)
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We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver complexities for all K × 2 × 2 × 2 tables.
Toric fiber products
, 2006
"... We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Gröbner bases f ..."
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Cited by 28 (5 self)
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We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Gröbner bases for these ideals. This allows us to unify and generalize some results in algebraic statistics.
Algebraic Statistical Models
 Statistica Sinica
, 2007
"... Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice propert ..."
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Cited by 19 (4 self)
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Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an 'algebraic exponential family'. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models.
Markov Bases of Binary Graph Models
, 2008
"... This paper is concerned with the topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We describe a degree four Markov basis for the model when the underlying graph is a cycle and generalize this resul ..."
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Cited by 14 (7 self)
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This paper is concerned with the topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We describe a degree four Markov basis for the model when the underlying graph is a cycle and generalize this result to the complete bipartite graph K2,n. We also give a combinatorial classification of degree two and three Markov basis moves as well as a Buchbergerfree algorithm to compute moves of arbitrary given degree. Finally, we compute the algebraic degree of the model when the underlying graph is a forest.
Invariant minimal Markov basis for sampling contingency tables with fixed marginals
, 2003
"... In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of invariant minimal Markov ..."
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Cited by 10 (6 self)
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In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of invariant minimal Markov basis. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (1998). Their algorithm is based on computing Gröbner basis of a wellspecified polynomial ideal. However the reduced Gröbner basis depends on the particular term order and is not symmetric. Therefore it is of interest to consider properties of invariant Markov basis. We study minimality of invariant Markov basis using techniques of Takemura and Aoki (2003).
Lattice Points, Contingency Tables, and Sampling
 In Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization
, 2004
"... Markov chains and sequential importance sampling (SIS) are described as two leading sampling methods for Monte Carlo computations in exact conditional inference on discrete data in contingency tables. Examples are explained from genotype data analysis, graphical models, and logistic regression. ..."
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Cited by 9 (2 self)
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Markov chains and sequential importance sampling (SIS) are described as two leading sampling methods for Monte Carlo computations in exact conditional inference on discrete data in contingency tables. Examples are explained from genotype data analysis, graphical models, and logistic regression.
Multigraded commutative algebra of graph decompositions
, 2012
"... The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe ..."
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Cited by 8 (3 self)
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The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in nonzero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K4minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.