Primary Decomposition in Singular

Macaulay is a system for computing in algebraic geometry and cummutative algebra; it is capable of a variety of computations which are tedious or impossible to perform by hand. The primitive types in the system are polynomials, matrices, ideals, polynomial rings, modules, maps between rings, and complexes of modules. The system performs algebraic manipulation on objects of these types. The possible manipulations include the computation of standard (Gröbner) bases, modules of syzygies, finite free resolutions, Hilbert polynomials and functions. Using these basic operations, a variety of derived operations are possible, such as projections, ideal intersections, and the computation of coherent sheaf cohomology groups. The algorithm used for constructing standard (Gröbner) bases and syzygies is described in ([Buc76], [Zac78], [Sch80], [Bay82], [BaSt86a]).

In the 2012–13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs. Volume 2 focuses on the most recent research.

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

Computation in commutative algebra and algebraic geometry has taken a major step forward in the last decade. This is due to the development of Grobner bases, computer algebra systems which are readily available to researchers, and the great increase in power of computers in this last decade.

Special issue on effective methods in algebraic computation

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

This thesis contains the classification of two-parameter bifurcations up to codimension three, using a two-parameter version of parametrised contact equivalence. Part one contains the classification up to codimension one. The result consists of the following components: 1. A list of normal forms for the germs having codimension less or equal to one. 2. Recognition conditions for each normal form in the list, i. e. conditions that characterise the equivalence class of the normal form. These conditions are equations and inequalities for the Taylor coefficients of the germs. 3. Universal unfoldings for each normal form. The result is obtained by investigating the structure of the orbits, which are induced by the action of the group of equivalences on the space of all bifurcation problems. Techniques from algebra, algebraic geometry and singularity theory are applied. In part two the classification is extended to codimension three. The second chapter of part two contains a generalisation of the singularity approach to equivariant bifurcation theory. The case of an action of a compact Lie group on state and parameter space is considered. The main example is the case of bifurcations with a certain D4-symmetry.

Although computer algebra is a research field in its own, its main driving force comes from various fields of applications. These applications range from mathematics and computer science over physics and engineering up to applications to technical and industrial problems.KeywordsModulus SpaceAlgebraic GeometryComputer AlgebraPolynomial SystemPrimary DecompositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahler measure of the first non-zero Alexander polynomial of the corresponding modules. We use the notion of pseudo-isomorphism, and also tools from commutative algebra and algebraic geometry, to reduce the conjectures to the case of torsion modules. We also describe concrete sequences which give the expected values of the limits in both cases. For this part we utilize a result of Bombieri and Zannier (conjectured before by A. Schinzel) and a result of Lawton (conjectured before by D. Boyd).

To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory-one of the two main sources of the modern discipline of commutative algebra.' The other source was algebraic geometry. It was in the setting of these two subjects that many of the main concepts and results of commutative algebra evolved in the 19th century. Thus commutative algebra can be said to have been well developed before it was created. To set the scene for our story, a word about mathematics, and especially algebra, in the 19th century. The period witnessed! fundamental transformations in mathematics -in its concepts, its methods, and in mathematicians' attitude toward their subject. These included a growing insistence on rigor and abstraction; a predisposition for founding general theories rather than focusing on specific problems; an acceptance of nonconstructive existence proofs; the emergence of mathematical specialties and specialists; the rise of the view of mathematics as a free human activity, neither deriving from, nor dependent on, nor necessarily applicable to concrete setting; the rise of set-theoretic thinking; and the reemergence of (a new variant of) the axiomatic method. As for algebra, its subject-matter and methods changed beyond recognition. Algebra as the study of solvability of polynomial equations gave way to algebra as the study of abstract structures defined axiomatically. While in the past algebra was on the periphery of mathematics, it now (the 19th and early 20th centuries) became one of its central concerns. Moreover, algebra began to penetrate other mathematical fields (e.g. geometry, analysis, logic, topology, number theory) to such an extent that in the early decades of the 20th century one began to speak of the algebraization of mathematics ([3, p. 135] and [22]).

While computer algebra systems have dealt with polynomials and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent problem. In this paper we explain how one system solves this problem, what types and operators it is necessary to introduce and, in short, how one can construct a computational theory of commutative algebra. Of necessity, such a theory is rather different from the conventional, non-constructive, theory. It is also somewhat different from the theories of Seidenberg [1974] and his school, who are not particularly concerned with practical questions of efficiency.KeywordsIntegral DomainCommutative AlgebraComputer Algebra SystemUnique FactorisationGreat Common DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The “peak reduction” method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was originally created by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well.In this paper, we present what seems to be the first applications of this technique in commutative algebra and affine algebraic geometry. We contribute toward a classification of two-variable polynomials by classifying, up to an automorphism, polynomials of the form \(a{\chi ^n} + b{y^m} + \sum\nolimits_{im + jn \leqslant mn} {cij{\chi ^i}} {y^j}\) (i.e., polynomials whose Newton polygon is either a triangle or a line segment). This has several applications to the study of embeddings of algebraic curves in the plane. In particular, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one can decide “almost” just by inspection whether or not a polynomial fiber {p(x, y) = 0} is an irreducible simply connected curve.Another application that we present here, yields a new decomposition of the group Aut(K(x, y)) in a free product with amalgamation.

The aim of this workshop was to communicate the most current developments in the field of noncommutative algebra and its interactions with algebraic geometry and representation theory.