Michael Stillman

Computational methods in commutative algebra and algebraic geometry

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.

On the complexity of computing syzygies

We give a self-contained exposition of Mayr & Meyers example of a polynomial ideal exhibiting double exponential degrees for the ideal membership problem, and generalise this example to exhibit minimal syzygies of double exponential degree. This demonstrates the existence of subschemes of projective space of double exponential regularity.

A theorem on refining division orders by the reverse lexicographic order

Let k be an infinite field of any characteristic, and let S = k[xl9..., xn] be a graded polynomial ring, where each xf has degree one. Let / c S be a homogeneous ideal. Let Sd denote the finite vector space of all homogeneous, degree d polynomials in S, so S = SQ

Computation of Hilbert functions

Abstract We present an algorithm along with implementation details and timing data for computing the Hilbert function of a monomial ideal. Our algorithm is often substantially faster in practice than existing algorithms, and executes in linear time when applied to an initial monomial ideal in generic coordinates. The algorithm generalizes to compute multi-graded Hilbert functions.

Cohomology on toric varieties and local cohomology with monomial supports

We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Coxs description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of rank one torsionfree sheaves on toric varieties.

Computations in algebraic geometry with Macaulay 2

This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. These expositions will be valuable to both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. The first part of the book is primarily concerned with introducing Macaulay2, whereas the second part emphasizes the mathematics.

Local cohomology of bivariate splines

We consider the problem of determining the dimension of the space of bivariate splines Ckr(Δ), for all k. This problem is closely related to the question of whether Cr(\gD) is a free R-module. The main result is that Cr(\gD) is free if and only if ¦Δ¦ has genus zero and Ckr(Δ) has the expected dimension for k = r + 1 (and hence for all k). We also obtain several interesting corollaries, including the following simple non-freeness criterion: given a fixed Δ having an edge with both vertices interior, and which does not extend to the boundary, there exists an r0, which can be determined by inspection, such that Cr(\gD) is not free for any r ≥ r0.

TORIC HILBERT SCHEMES

We introduce and study the toric Hilbert scheme that parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal.

Parameter estimation for Boolean models of biological networks

Boolean networks have long been used as models of molecular networks, and they play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.

Strategies for Computing Minimal Free Resolutions

In the present paper we study algorithms based on the theory of Grobner bases for computing free resolutions of modules over polynomial rings. We propose a technique which consists in the application of special selection strategies to the Schreyer algorithm. The resulting algorithm is efficient and, in the graded case, allows a straightforward minimalization algorithm. These techniques generalize to factor rings, skew commutative rings, and some non-commutative rings. Finally, the proposed approach is compared with other algorithms by means of an implementation developed in the new system Macaulay2.