Root polytopes and growth series of root lattices (2008)
Ardila, Federico, Beck, Matthias, Hosten, Serkan, Pfeifle, Julian, Seashore, Kim
The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root...
Least Squares Methods for Equidistant Tree Reconstruction (2008)
Fahey, Conor, Hosten, Serkan, Krieger, Nathan, Timpe, Leslie
UPGMA is a heuristic method identifying the least squares equidistant phylogenetic tree given empirical distance data among $n$ taxa. We study this classic algorithm using the geometry of the space...
Normal Toric Ideals of Low Codimension (2008)
Dueck, Pierre, Hosten, Serkan, Sturmfels, Bernd
Every normal toric ideal of codimension two is minimally generated by a Grobner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of...
Supernormal vector configurations (2007)
Serkan Hosten, Diane Maclagan, Bernd Sturmfels, Let B
Abstract. A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including...
Karin Gatermann, Serkan Hosten
Abstract: We provide algorithmic methods for the solution of the classication problem of bifurcations by vanishing and non-vanishing derivatives. These methods come from a generalization of standard...
Serkan Hosten, Francisco Santos, Bernd Sturmfels, Communicated Gunter, M. Ziegler
Abstract. We study the convex hull PA of the 0-1 incidence vectors of all triangulations of a point configuration A. This was called the universal polytope in [4]. The affine span of PA is described...
Serkan Hosten, Bernd Sturmfels
Integer programming (IP) is concerned with solving linear equations over the non-negative integers, subject to the requirement that a linear cost function is minimized. We consider families of...
On the Complexity of Smooth Projective Toric Varieties (2007)
In this paper we answer a question posed by Batyrev, which asks if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth projective toric...
The Algebraic Complexity of Maximum Likelihood Estimation for Bivariate Missing Data (2007)
Hosten, Serkan, Sullivant, Seth
We study the problem of maximum likelihood estimation for general patterns of bivariate missing data for normal and multinomial random variables, under the assumption that the data is missing at...
Buot, Max-Louis G., Hosten, Serkan, Richards, Donald St. P.
Let $\mu$ be a $p$-dimensional vector, and let $\Sigma_1$ and $\Sigma_2$ be $p \times p$ positive definite covariance matrices. On being given random samples of sizes $N_1$ and $N_2$ from independent...
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,...
Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices (2005)
Beck, Matthias, Hosten, Serkan
The coordination sequence of a lattice $\L$ encodes the word-length function with respect to $M$, a set that generates $\L$ as a monoid. We investigate the coordination sequence of the cyclotomic...
Solving the Likelihood Equations (2004)
Hosten, Serkan, Khetan, Amit, Sturmfels, Bernd
Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of...
The Maximum Likelihood Degree (2004)
Catanese, Fabrizio, Hosten, Serkan, Khetan, Amit, Sturmfels, Bernd
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization...
A finiteness theorem for Markov bases of hierarchical models (2004)
Hosten, Serkan, Sullivant, Seth
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 beyond a...
Ideals of Adjacent Minors (2004)
We give a description of the minimal primes of the ideal generated by the 22 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent mm minors...
Ideals of Adjacent Minors (2003)
Hosten, Serkan, Sullivant, Seth
We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m...
Computing the Integer Programming Gap (2003)
Hosten, Serkan, Sturmfels, Bernd
We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is...
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...
Supernormal Vector Configurations (2001)
Hosten, Serkan, Maclagan, Diane, Sturmfels, Bernd
A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including triangulations,...
The vertex ideal of a lattice (2000)
Hosten, Serkan, Maclagan, Diane
We introduce a monomial ideal whose standard monomials encode the vertices of all fibers of a lattice. We study the minimal generators, the radical, the associated primes and the primary...
Primary Decomposition of Lattice Basis Ideals (2000)
We study the primary decomposition of lattice basis ideals. These ideals are binomial ideals with generators given by the elements of a basis of a saturated integer lattice. We show that the minimal...
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...
The order dimension of the complete graph (1999)
Serkan Hosten, Walter D. Morris
We show that the order dimension of the complete graph on n vertices is the smallest integer t for which there are n antichains in the subset lattice of [t \Gamma 1] that do not contain [t \Gamma 1]...
SAGBI and SAGBI-Grobner bases over principal ideal domains (1999)
William W. Adams, Serkan Hosten, Philippe Loustaunau, J. Lyn Miller
In this paper we will discuss computational and structural properties of subalgebras of polynomial rings when the base ring is a principal ideal domain (PID). The objects we study are the so-called...
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...
Sagbi And Sagbi-Gröbner Bases Over Principal Ideal Domains (1998)
William W. Adams, Serkan Hosten, Philippe Loustaunau, J. Lyn Miller
this paper we will discuss computational and structural properties of subalgebras of polynomial rings when the base ring is a principal ideal domain (PID). The objects we study are the so-called...
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...
On the Complexity of Smooth Projective Toric Varieties (1996)
In this paper we answer a question posed by V.V. Batyrev. The question asked if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth...
The Polytope of All Triangulations of a Point Configuration (1996)
Serkan Hosten, Francisco Santos, Bernd Sturmfels
We study the convex hull PA of the 0-1 incidence vectors of all triangulations of a point configuration A. This was called the universal polytope in [4]. The affine span of PA is described in terms...