Gert-Martin W. Greuel

RPTU - Rheinland-Pfälzische Technische Universität Kaiserslautern Landau | TUK · Algebra, Geometry and Computer algebra Group

We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of...

In 2011, Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a certain equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by constructing lists of normal forms. The...

We present new results on standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring over a computable field K . We prove the semicontinuity of the “highest corner” in a family of ideals, parametrized by the spectrum of a Noetherian domain A . This semicontinuity is used to design a new...

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action a...

We present new results and an algorithm for standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring in finitely many variables over a field K. The algorithm provides a significant speed up if K is the quotient field of a Noetherian integral domain A, when coefficient swell occurs. Th...

We consider families of schemes over arbitrary fields resp. analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular families of generically reduced curves. We define a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points and prove...

We give a brief historical overview of the development of the Computer Algebra System SINGULAR: why it came about and how the development was related to the attempt to refute Zariski's multiplicity conjecture. To be published as an appendix to the article "Topological Equisingularity: Old Problems From a new Perspective" by J. Fernandez De Bobadil...

We give a brief historical overview of the development of the Computer Algebra System SINGULAR: why it came about and how the development was related to the attempt to refute Zariski's multiplicity conjecture.

We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of...

Crystals have always fascinated people. Not only their mysterious variety of colour refraction, but also their regular geometry and special symmetry surprise and delight repeatedly. This article will briefly outline the history of crystallography from the Platonic solids to quasicrystals and the mathematical-geometrical description of crystal struc...

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal power series parametrized by some algebraic variety. In contrast to the case of analytic or algebraic families...

We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$...

A historical outline of the interaction between two disciplines. My essay is a historical survey on crystals and mathematics. Although most of the material is folklore, I gave rigorous references and perhaps new views: - The relation of the Greek's view of the platonic solids to modern mathematics. - The relation of the crystals of a snowflake to...

We consider the actions of different groups G on the space M of m x n matrices with entries in the formal power series ring K[[x1,..., xs]], K an arbitrary field. G acts on M by analytic change of coordinates, combined with the multiplication by invertible matrices from the left, the right or from both sides, respectively. This includes right and c...

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal power series parametrized by some algebraic variety. In contrast to the case of analytic or algebraic families...

Let $R=K[[x_1,...,x_s]]$ be the ring of formal power series with maximal ideal $\mathfrak{m}$ over a field $K$ of arbitrary characteristic. On the ring $M_{m,n}$ of $m\times n$ matrices $A$ with entries in $R$ we consider several equivalence relations given by the action on $M_{m,n}$ of a group $G$. $G$ can be the group of automorphisms of $R$, com...

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x 1 ,…,x s ]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim(R/I)>0 and K is an infinite field (of arbitrary characteristic). Here two ideals I...

We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different invariants, such as the Milnor number, the Tjurina number, and the dimension of a smoothing component.

We describe different approaches to prove \(H^1\)-vanishing for ideal sheaves of zero-dimensional schemes. When looking for appropriate \(H^1\)-vanishing theorems for the problems discussed in Chap. 4, one has to be aware that the types of zero-dimensional schemes to be considered are quite different (cf. Sects. 2.2.1.4 and 2.3.4).

Let $R=K[[x_1,...,x_s]]$ be the ring of formal power series with maximal ideal $\mathfrak{m}$ over a field $K$ of arbitrary characteristic. On the ring $M_{m,n}$ of $m\times n$ matrices $A$ with entries in $R$ we consider several equivalence relations given by the action on $M_{m,n}$ of a group $G$. $G$ can be the group of automorphisms of $R$, com...

Let $R=K[[x_1,...,x_s]]$ be the ring of formal power series with maximal ideal $\mathfrak{m}$ over a field $K$ of arbitrary characteristic. On the ring $M_{m,n}$ of $m\times n$ matrices $A$ with entries in $R$ we consider several equivalence relations given by the action on $M_{m,n}$ of a group $G$. $G$ can be the group of automorphisms of $R$, com...

We study "straight equisingular deformations", a linear subfunctor of all equisingular deformations and describe their seminuniversal deformation by an ideal containing the fixed Tjurina ideal. Moreover, we show that the base space of the seminuniversal straight equisingular deformation appears as the fibre of a morphism from the {\mu}-constant str...

In this survey paper we give an overview on some aspects of singularities of algebraic varieties over an algebraically closed field of arbitrary characteristic. We review in particular results on equisingularity of plane curve singularities, classification of hypersurface singularities and determinacy of arbitrary singularities. The section on equi...

This volume brings together recent, original research and survey articles by leading experts in several fields that include singularity theory, algebraic geometry and commutative algebra. The motivation for this collection comes from the wide-ranging research of the distinguished mathematician, Antonio Campillo, in these and related fields. Besides...

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action a...

This section is devoted to the study of zero-dimensional schemes in a smooth projective surface \(\varSigma \), associated to and concentrated in the (finite) set of singular points of a reduced curve C on \(\varSigma \). In this chapter, a curve (singularity) we will always mean a reduced curve (singularity), unless we explicitly say the opposite....

We are ready to accomplish our main task, that is to answer the two following questions concerning equisingular families (ESF) of curves. whether a family of algebraic curves with a prescribed collection of singularities form a nonempty, T-smooth (i.e. smooth of expected dimension), irreducible stratum in the discriminant in a given linear system |...

Global deformation theory serves as a key tool in the study of families of singular algebraic varieties, notably, equisingular families of algebraic curves, the main object of this monograph.

From the Preface: The title Singular algebraic curves designates a wide research field which has been of constant interest and importance from Descartes, Pascal, Newton to nowadays. We do not pursue the “mission impossible” to give a complete account of this topic, but concentrate mainly on the geometry of deformations and families of singular alg...

Egbert Brieskorn died on July 11, 2013, a few days after his 77th birthday. He was an impressive personality who has left a lasting impression on all who knew him, whether inside or outside of mathematics. Brieskorn was a great mathematician, but his interests, his knowledge, and activities ranged far beyond mathematics. In this contribution, which...

In this survey paper we give an overview on some aspects of singularities of algebraic varieties over an algebraically closed field of arbitrary characteristic. We review in particular results on equisingularity of plane curve singularities, classification of hypersurface singularities and determinacy of arbitrary singularities. The section on equi...

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $\mu$ and the Tjurina number $\tau$ of any isolated plane curve s...

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity, in particular of a complete intersection, and use them to compare the Milnor number, the Tjurina number and the dimension of the torsion part of the 1-forms.

Let M be the ring of m x n matrices A with entries in R=K[[x1,...,xs]], the ring of formal power series over an arbitrary field K. We call A finitely determined if any matrix B, with entries of A-B in <x1,...,xs>^k for some k, is left-right equivalent to A, i.e. B is contained in the G-orbit of A, where G is the group of automorphisms of R combined...

The well-known Mather-Yau theorem says that the isomorphism type of the local ring of an isolated complex hypersurface singularity is determined by its Tjurina algebra. It is also well known that this result is wrong as stated for power series f in K[[x]] over fields K of positive characteristic. In this note we show that, however, also in positive...

We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using $\delta$ and $\mu$. Mo...

Egbert Brieskorn died on July 11, 2013, a few days after his 77th birthday. He was an impressive personality who has left a lasting impression on all who knew him, whether inside or outside of mathematics. Brieskorn was a great mathematician, but his interests, his knowledge, and activities ranged far beyond mathematics. In this contribution, which...

swMATH is a novel information service for mathematical software. It offers open access to a comprehensive database with information on mathematical software and provides a systematic collection of references and linking to software-relevant mathematical publications.

The well-known Mather-Yau theorem says that the isomorphism type of the local ring of an isolated complex hypersurface singularity is determined by its Tjurina algebra. It is also well known that this result is wrong as stated for power series f in K[[x]] over fields K of positive characteristic. In this note we show that, however, also in positive...

An information service for mathematical software is presented. Publications and software are two closely connected facets of mathematical knowledge. This relation can be used to identify mathematical software and find relevant information about it. The approach and the state of the art of the information service are described here.

We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities {f (x, y) = 0} and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number resp. the delta-invariant δ can be computed by explicit form...

We classify isolated hypersurface singularities $f\in K[[x_1,..., x_n]]$, $K$ an algebraically closed field of characteristic $p>0$, which are simple w.r.t. right equivalence, that is, which have no moduli up to analytic coordinate change. For $K=\mathbb R$ or $\mathbb C$ this classification was initiated by Arnol'd, resulting in the famous ADE-ser...

Possibly more than any other science, mathematics of today finds itself between the conflicting demands of research, application, and communication. A great part of modern mathematics regards itself as searching for inner mathematical structures just for their own sake, only committed to its own axioms and logical conclusions. To do so, neither ass...

This paper presents a new SMT solver, STABLE, for formulas of the quantifier-free logic over fixed-sized bit vectors (QF-BV). The heart of STABLE is a computer-algebra-based engine which provides algorithms for simplifying arithmetic problems of an SMT instance prior to bit-blasting. As the primary application domain for STABLE we target an SMT-bas...

We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities $\{f(x,y) = 0\}$ and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number $\mu$ resp. the delta-invariant $\delta$ can be computed b...

In connection with his classification of real and complex hypersurface singularities Arnol'd introduced in the 1970's the condition A which allows to compute a normal form of a power series f with respect to right equivalence. For this he uses piecewise filtrations induced by the Newton polytope of f. Wall considered in 1999 a non-degeneracy condit...

We study singularities f in K[[x_1,...,x_n]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positiv...

We present foundational work on standard bases over rings and on Boolean Gröbner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems...

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. A modification, applicable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper has been implemented in Singular, for localizations of affine ri...

In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our...

An abstract is not available.

We propose a new approach for static translinear network synthesis. We model the network topology in terms of graph theory, leading to a catalog of valid translinear networks. The catalog serves as a synthesis tool for circuits with given desired input-output behavior. Methods from algebraic geometry and computer algebra are used to match the desir...

This paper proposes a new approach for proving arithmetic correctness of data paths in System-on-Chip modules. It complements existing techniques which are, for reasons of complexity, restricted to verifying only the control behavior. The circuit is modeled at the arithmetic bit level (ABL) so that our approach is well adapted to current industrial...

The Hilbert function of a graded module associates to an integer n the dimension of the n-th graded part of the given module. For sufficiently large n, the values of this function are given by a polynomial, the Hilbert polynomial. To show this, we use the Hilbert-Poincaré series, a formal power series in t with coefficients being the values of the...

It is well-known that every integer is a product of prime numbers, for instance 10 = 2.5. This equation can also be written as an equality of ideals, 〈10〉 =〈2〉 ∩ 〈5〉 in the ring ℤ. The aim of this section is to generalize this fact to ideals in arbitrary Noetherian rings.

Integral extension of a ring means adjoining roots of monic polynomials over the ring. This is an important tool for studying affine rings, and it is used in many places, for example, in dimension theory, ring normalization and primary decomposition. Integral extensions are closely related to finite maps which, geometrically, can be thought of as p...

For certain applications the local rings K[x]〈x〉, x = (x1, ... , xn ), are not “sufficiently local”. As explained in Appendix A, Sections A.8 and A.9, the latter rings contain informations about arbitrary small Zariski neighbourhoods of 0 ∈ Kn . Such neighbourhoods turn out to be still quite large, for instance, if n = 1 then they consist of K minu...

The concept of a ring is probably the most basic one in commutative and non-commutative algebra. Best known are the ring of integers ℤ and the polynomial ring K[x] in one variable x over a field K.

From the reviews: "...It is certainly no exaggeration to say that Greuel and Pfister's A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra, in which computational methods and results become central to how the subject is taught and learned. [...] Among the great strengths...

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis i...

We study several deformation functors associated to the normalization of a reduced curve singularity (X, 0) Ì (\mathbbCn, 0){(X, 0) \subset (\mathbb{C}^n, 0)} . The main new results are explicit formulas, in terms of classical invariants of (X, 0), for the cotangent cohomology groups T i , i = 0,1,2, of these functors. Thus we obtain precise sta...

Jon Kleinberg's work perfectly fits the Nevanlinna Prize specification since, as we have seen, his mathematical insights have had wide application to multiple elements of information science-the effectiveness of advanced Web search engines, Internet routing, data mining, and the sociology of the World Wide Web. We refer the interested reader to his...

In this paper we develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniveral deformation is smooth in both cases. Our approach through deformations of the parametrization is ele...

In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in numerical, universal and asymptotically proper sufficient conditions to guarantee the nonemptyness, T-smoothness and irreducibility of the variety of all pr...

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences. This book presents the basic s...

We study deformations of germs of reduced complex curve singularities and of singular projective curves in some Pn(ℂ). In both cases a deformation is topologically trivial iff the Milnor numbers of the singularities are constant during the deformation. The Milnor number also occurs naturally in the degree of the singular Todd class of Baum-Fulton-M...

We show that analytic solutions of the Ernst equation with a non-empty zero-level set of lead to smooth ergosurfaces in spacetime. In fact, the spacetime metric is smooth near an 'Ernst ergosurface' Ef if, and only if, is smooth near Ef and does not have zeros of infinite order there.

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main result states that a finite group G is solvable if and only if for some n the identity $u_n(x,y)\equiv 1$ hol...

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of...

We explain how the computer algebra system SINGULAR has been used successfully to either solve a mathematical problem or to find the correct statement of a theorem which then could be proved without computer, or to construct interesting examples. The applications belong to algebraic geometry and singularity theory, the main area of applications, bu...

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