This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Computational methods in commutative algebra and algebraic geometry

Let G be a finite graph on [n] = {1, 2,…, n}, X a 2 × n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this article, we study about ideals I G of S generated by 2-minors [i, j] of X which correspond to edges {i, j} of G. In particular, we construct a Grobner basis of I G as a set of paths of G and compute a primary decomposition.

Graphs and Ideals Generated by Some 2-Minors

Polynomial rings and ideals Grobner bases First applications Grobner bases for modules Grobner bases of toric ideals Selected applications in commutative algebra and combinatorics Bibliography Index

Grobner Bases in Commutative Algebra

Researching Your Own Practice

We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

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Cohen-Macaulay binomial edge ideals

Binomial edge ideals and conditional independence statements

We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gr\"obner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. 
Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation

A Case Where Choosing a Product Order Makes the Calculations of a Groebner Basis Much Faster

Let X={xij} and Y={yij} be generic n by n matrices and Z = XY –YX. Let where k is a field with char K ≠ 2 and let I be the ideal generated by the entries of Z. Denote by R the quotient ring S/I. In this paper we study its Koszul dual, which is the algebra generated by and denoted by R !, and show that for n≥3 it is the enveloping algebra of a nilpotent Lie algebra.

The koszul dual of the ring of commuting matrices

Let $X=(x_{ij})$ and $Y=(y_{ij})$ be generic $n$ by $n$ matrices and $Z=XY-YX$. Let $S=k[x_{11},...,x_{nn},y_{11},...,y_{nn}]$, where $k$ is a field, let $I$ be the ideal generated by the entries of $Z$ and let $R=S/I$. We give a conjecture on the first syzygies of $I$, show how these can be used to give a conjecture on the canonical module of $R$. Using this and the Hilbert series of $I$ we give a conjecture on the Betti numbers of $I$ in the $4 \times 4$ case. We also give some guesses on the structure of the resolution in general.

Freyja Hreinsdóttir

Papers

Binomial edge ideals and conditional independence statements

Binomial edge ideals and conditional independence statements

A Case Where Choosing a Product Order Makes the Calculations of a Groebner Basis Much Faster

The koszul dual of the ring of commuting matrices

Conjectures on the ring of commuting matrices