Theta Bodies for Polynomial Ideals (2008)
Gouveia, João, Parrilo, Pablo A., Thomas, Rekha R.
Inspired by a question of Lov\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the...
Lectures in Geometric Combinatorics (2008)
DMS-0401047 from the National Science Foundation. Abstract. The fourteen lectures in this book were prepared for the advanced undergraduate course at the Park City Mathematics Institute on Geometric...
Computational algebra and combinatorics of toric ideals (2008)
Diane Maclagan, Rekha R. Thomas, Sara Faridi, Leah Gold, A. V. Jayanthan, Amit Khetan, ...
Moduli of McKay quiver representations I: the coherent component (2007)
Craw, Alastair, Maclagan, Diane, Thomas, Rekha R.
For a finite abelian group G ⊂ GL (n, \[k\]), we describe the coherent component Yθ of the moduli space ℳθ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric...
Bogart, Tristram, Thomas, Rekha R.
We introduce a new measure of complexity of integer hulls of rational polyhedra called the small Chvatal rank (SCR). The SCR of an integer matrix A is the number of rounds of a Hilbert basis...
Moduli of McKay quiver representations II: Groebner basis techniques (2006)
Craw, Alastair, Maclagan, Diane, Thomas, Rekha R.
In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Groebner bases and toric geometry. For a finite abelian group...
Fukuda, Komei, Jensen, Anders N., Thomas, Rekha R.
This paper presents algorithms for computing the Gr¨obner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Gröbner bases of the ideal. Our algorithms are...
Tristram Bogart, Tristram Bogart, Rekha R. Thomas, Rekha R. Thomas, Douglas A. Lind, Isabella Novik, ...
This is to certify that I have examined this copy of a doctoral dissertation by
Nice Initial Complexes of Some Classical Ideals (2005)
Conca, Aldo, Hosten, Serkan, Thomas, Rekha R.
This is a survey article on Gorenstein initial complexes of extensively studied ideals in commutative algebra and algebraic geometry. These include defining ideals of Segre and Veronese varieties,...
Computing Groebner Fans (2005)
Fukuda, Komei, Jensen, Anders N., Thomas, Rekha R.
This paper presents algorithms for computing the Groebner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Groebner bases of the ideal. Our algorithms are...
The Circuit Ideal of a Vector Configuration (2005)
Bogart, Tristram, Jensen, Anders N., Thomas, Rekha R.
The circuit ideal, $\ica$, of a configuration $\A = \{\a_1, ..., \a_n\} \subset \Z^d$ is the ideal generated by the binomials ${\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n]$ as $\cc = \cc^+ -...
Reverse Lexicographic and Lexicographic Shifting (2005)
Babson, Eric, Novik, Isabella, Thomas, Rekha R.
A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, $\Delta_{\lex}$ -- an operation that transforms a...
Moduli of McKay quiver representations I: the coherent component (2005)
Craw, Alastair, Maclagan, Diane, Thomas, Rekha R.
For a finite abelian group G in GL(n,k), we describe the coherent component Y_theta of the moduli space M_theta of theta-stable McKay quiver representations. This is a not-necessarily-normal toric...
Toric Initial Ideals of $\Delta$-Normal Configurations: Cohen-Macaulayness and Degree Bounds (2003)
O'Shea, Edwin, Thomas, Rekha R.
A normal (respectively, graded normal) vector configuration $A$ defines the toric ideal $I_A$ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when...
The toric Hilbert scheme of a rank two lattice is smooth and irreducible (2002)
Maclagan, Diane, Thomas, Rekha R.
The toric Hilbert scheme of a lattice L in Z^n is the multigraded Hilbert scheme parameterizing all ideals in k[x_1,...,x_n] with Hilbert function value one for every degree in the grading monoid...
An Algebraic Perspective of Group Relaxations (2001)
This is an expository article on recent developments in the theory of group relaxations in integer programming from an algebraic perspective.
Gomory Integer Programs (2001)
Hosten, Serkan, Thomas, Rekha R.
The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but...
Algorithms for the Toric Hilbert Scheme (2000)
Stillman, Michael, Sturmfels, Bernd, Thomas, Rekha R.
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a...
Computing Gröbner fans of toric ideals (2000)
Huber, Birkett, Thomas, Rekha R.
The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Gröbner fan of the ideal is the normal...
Combinatorics of the toric Hilbert scheme (1999)
Maclagan, Diane, Thomas, Rekha R.
The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric...
Standard pairs and group relaxations in integer programming (1999)
Serkan Hosten, Rekha R. Thomas
The main result of this paper is a non-Buchberger algorithm for constructing initial ideals and Grobner bases of toric ideals, based on the connections between toric ideals and integer programming....
The associated primes of initial ideals of lattice ideals (1999)
Serkan Hosten, Rekha R. Thomas
This paper concerns the associated primes and primary decompositions of the monomial initial ideals of lattice ideals. For a fixed initial ideal, we show that the multiplicities of its associated...
Combinatorics Of The Toric Hilbert Scheme (1999)
Diane Maclagan, Rekha R. Thomas, R. Thomas
. The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric...
The Associated Primes of Initial Ideals of Lattice Ideals (1999)
Serkan Hosten, Rekha R. Thomas
This paper concerns the associated primes and primary decompositions of the monomial initial ideals of lattice ideals. For a fixed initial ideal, we show that the multiplicities of its associated...
Algebraic Methods In Integer Programming (1999)
This article highlights some of the recent results in theoretical integer programming that have been obtained by studying
Computing Gröbner Fans of Toric Ideals (1999)
Birkett Huber, Rekha R. Thomas
The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Grobner fan of the ideal is the normal...
Standard Pairs and Group Relaxations in Integer Programming (1998)
The main result of this paper is a non-Buchberger algorithm for constructing initial ideals and Grobner bases of toric ideals, based on the connections between toric ideals and integer programming....
Gröbner Bases in Integer Programming (1998)
this paper we fix a d by n integral matrix A of full row rank. Let ker(A) denote the n \Gamma d dimensional subspace fu 2 R
Truncated gröbner bases for integer programming (1997)
Rekha R. Thomas, Robert Weismantel
In this paper we introduce a multivariate grading of the toric ideal associated with the
Applications To Integer Programming (1997)
. Integer programs are a class of optimization problems that have traditionally been studied using techniques from polyhedral theory and linear algebra. Although these methods have proved to be very...
SPIN: A Software for Computing State Polytopes of Toric Ideals (1997)
Birkett Huber, Rekha R. Thomas
this paper we present a software package for computing the state polytope of a toric ideal based on theory and algorithms presented in [10]. Given an integer matrix A 2 Z
Gröbner Bases and Integer Programming (1997)
Serkan Hosten, Rekha R. Thomas
This article is a brief survey of recent work on Gröbner bases (Buchberger 1965) of toric ideals and their role in integer programming. Toric varieties and ideals are crucial players in the...
Truncated Gröbner Bases for Integer Programming (1995)
Rekha R. Thomas, Robert Weismantel
The toric ideal I A of a matrix A = (a 1 ; : : : ; an ) 2 ZZ d\Thetan is the kernel of the monoid algebra map ßA : k[x 1 ; : : : ; xn ] ! k[t \Sigma1 1 ; : : : ; t \Sigma1 d ], defined as x j 7! t a...
A Geometric Buchberger Algorithm for Integer Programming (1995)
Let IP denote the family of integer programs of the form Min cx : Ax = b, x ∈ N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of...
An Algebraic Geometry Algorithm for Scheduling in Presence of Setups and Correlated Demands (1994)
Sridhar R. Tayur, Rekha R. Thomas, N. R. Natraj
We study here a problem of scheduling n job types on m parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance...
Variation of Cost Functions in Integer Programming (1994)
Bernd Sturmfels, Rekha R. Thomas
We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal...
Gröbner Bases And Triangulations Of The Second Hypersimplex (1994)
Bernd Sturmfels, Rekha R. Thomas
The algebraic technique of Grobner bases is applied to study triangulations of the second hypersimplex \Delta(2; n). We present a quadratic Grobner basis for the associated toric ideal I(Kn ). The...