Binomial Ideals

  title={Binomial Ideals},
  author={David Eisenbud and Bernd Sturmfels},
: We investigate the structure of ideals generated by binomials (poly-nomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and it has numerous applications within and beyond pure mathematics. The ideals defining toric varieties are precisely the binomial prime ideals. Our main results concern primary decomposition: If I is a binomial ideal then the radical, associated primes, and… 

Decompositions of binomial ideals

We present Binomials, a package for the computer algebra system Macaulay 2, which specializes well-known algorithms to binomial ideals. These come up frequently in algebraic statistics and

The height of binomial ideals and toric $K$-algebras with isolated singularity

. We give an upper bound for the height of an arbitrary binomial ideal I in terms of the dimension of a vector space spanned by integer vectors corresponding to a set of binomial generators of I .

Betti numbers of binomial ideals

Introduction to Binomial Ideals

In this chapter we introduce the main topic of this book: binomials and binomial ideals. Special attention is given to toric ideals. These are binomial ideals arising from an integer matrix which

Gröbner bases of binomial ideals associated with finite graphs and polyominoes

Binomial ideals appear in various areas of pure mathematics as well as of applied mathematics, including algebraic geometry, commutative algebra, combinatorics and algebraic statistics. In the

Binomial Edge Ideals and Related Ideals

In this chapter we consider classes of binomial ideals which are naturally attached to finite simple graphs. The first of these classes are the binomial edge ideals. These ideals may also be viewed

Finding binomials in polynomial ideals

An algorithm which finds binomials in a given ideal by reduction to the Artinian case using tropical geometry and in particular decides whether binomial exist in I at all.

Primary Decomposition of Lattice Basis Ideals

All minimal primes in the 3  ×  n case are determined, and faster ways of computing a generating set for the associated toric ideal from a lattice basis ideal are presented.

Restricted classes of veronese type ideals and algebras

This paper focuses on Veronese ideals of bounded support, which are ideals which are generated by monomials of degree $d$ in the polynomial ring in variables and which satisfy certain numerical side conditions regarding their exponents.

A Divide and Conquer Method to Compute Binomial Ideals

The divide and conquer strategy breaks the problem into subproblems in rings of lesser number of variables than the original ring and applies the framework on five problems – radical, saturation, cellular decomposition, minimal primes of binomial ideals, and computing a generating set of a toric ideal.



Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory

Problems connected with ideals generated by finite sets F of multivariate polynomials occur, as mathematical subproblems, in various branches of systems theory, see, for example, [5]. The method of

Direct methods for primary decomposition

SummaryLetI be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical ofI, for localizingI with respect to another ideal, and

Asymptotic analysis of toric ideals

This partially expository note extends and refines our earlier work on Grobner bases of toric varieties [14]. It is closely related to the theory of A-hypergeometric functions due to Gel′fand Graev,

A generalized Koszul complex. II. Depth and multiplicity

Introduction. In a previous paper [3], a general complex was described which could be associated with a map of modules over a commutative ring. Also mentioned in that paper were several areas which

A Geometric Buchberger Algorithm for Integer Programming

An algorithm for the construction of a unique minimal test set for this family of integer programs called the reduced Grobner basis of IP{A, c} is presented which is called a geometric Buchberger algorithm for integer programming and it is shown how an integer program may be solved using this test set.

Commutative Algebra: with a View Toward Algebraic Geometry

Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of