Papers by Gert-Martin Greuel
Cornell University - arXiv, Jan 5, 2021
We consider families of parameterizations of reduced curve singularities over a Noetherian base s... more 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 parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta and the semicontinuity theorem provides a simultaneous bound for the determinacy in a family. The fact that the base space can be an arbitrary Noetherian scheme causes some difficulties but is (not only) of interest for computational purposes.
Mathematics in Computer Science
We present new results on standard basis computations of a 0-dimensional ideal I in a power serie... more 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 modular algorithm for computing a standard basis of I if K is the quotient field of A. It uses the computation over the residue field of a “good” prime ideal of A to truncate high order terms in the subsequent computation over K. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps $$A={\mathbb {Z}}$$ A = Z and $$A=k[t]$$ A = k [ t ] , k any field and t a set of paramet...
Cornell University - arXiv, Nov 9, 2017
In this survey paper we give an overview on some aspects of singularities of algebraic varieties ... more 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 equisingularity has its roots in two important early papers by Antonio Campillo. One emphasis is on the differences between positive and zero characteristic and on open problems.
Cornell University - arXiv, Sep 3, 2017
We collect some classical results about holomorphic 1-forms of a reduced complex curve singularit... more 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. 1 Classical results This note was motivated by a recent preprint of A. Dimca [Di17] one 1forms of irreducible plane curve singularities. We generalize his results to not necessarily irreducible complete intersection curve singularities by deducing them from classical results (partly in German), which are somewhat scattered in the literature and apparently not well known. We collect them here with reference to the original sources. Consider a reduced complex curve singularity (X, 0) ⊂ (C n , 0) defined by an ideal I ⊂ O C n ,0 with r = r(X, 0) branches. Let n : (X,0) → X, 0) be the normalization, where (X,0) is the multi-germ consisting of r smooth branches. We set
Cornell University - arXiv, Jul 14, 2021
We prove the semicontinuity of the delta invariant in a family of schemes or analytic varieties w... more We prove the semicontinuity of the delta invariant in a family of schemes or analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular for families of generically reduced curves. We define and use a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points. Our results generalize results by Teissier and Chiang-Hsieh-Lipman for families of reduced curve singularities. The base ring for our families can be an arbitrary PID such that our semicontinuity result provides possible improvements for algorithms to compute the genus of a curve.
This paper describes our new satisfyability (SAT) modulo theory (SMT) solver STABLE for the quant... more This paper describes our new satisfyability (SAT) modulo theory (SMT) solver STABLE for the quantifier-free logic over fixed size bit vectors. Our main application domain is formal verification of system-on-chip (SoC) modules designed for complex computational tasks, for example, in signal processing applications. Ensuring proper functional behavior for such modules, including arithmetic correctness of the data paths, is considered a very difficult problem. We show how methods from computer algebra can be integrated into an SMT solver such that instances can be handled where the arithmetic problem parts are specified mixing various levels of abstraction from the plain gate level for small highly optimized components up to the pure word level used in high-level specifications. If the arithmetic problem parts include multiplications such mixed problem descriptions quickly drive current SMT solvers towards their capacity limits. High performance data paths are often designed at a level...
Methods and Applications of Analysis, 2017
We present new results on equisingularity and equinormalizability of families with isolated non-n... more We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a δ-invariant and a µ-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using δ and µ. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families. Finally we state some open problems and conjectures. In addition we give a survey of classical results about equisingularity and equinormalizability so that the article may be useful as a reference source. 6 Simultaneous normalization 39 7 Families of isolated non-normal singularities 47 8 Connected components of the Milnor fibre 61 9 Topology of families of generically reduced curves 68 10 Comments, open problems and conjectures 75 10.1 Equisingularitiy for families of generically reduced curves. .. 75 10.2 Deformation of the normalization and of the parametrization 78
Mathematische Zeitschrift, 2022
Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free a... more Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.
Endbericht IMAGINARY : Wanderausstellung des Mathematischen Forschungsinstituts Oberwolfach ; Dezember 2007 - Dezember 2008 ; Wissenschaftsjahr 2008: Mathematik - Alles, was zählt ; [Übersicht der Wanderausstellung]
Springer Monographs in Mathematics, 2018
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Springer Monographs in Mathematics, 2007
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
A Panorama of Singularities, 2020
We study "straight equisingular deformations", a linear subfunctor of all equisingular deformatio... more 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 µ-constant stratum onto a punctual Hilbert scheme parametrizing certain zerodimensional schemes concentrated in the singular point. Although equisingular deformations of plane curve singularities are very well understood, we believe that this aspect may give a new insight in their inner structure.
Handbook of Geometry and Topology of Singularities I, 2020
We give a survey on some aspects of deformations of isolated singularities. In addition to the pr... more 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.
Global Deformation Theory
Springer Monographs in Mathematics, 2018
Global deformation theory serves as a key tool in the study of families of singular algebraic var... more 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.
Ali Nesin and the Nesin Mathematics Village
Proceedings of the International Congress of Mathematicians (ICM 2018), 2019
arXiv: Algebraic Geometry, 1999
We study families V of curves in P2(C) of degree d having exactly r singular points of given topo... more We study families V of curves in P2(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, i.e., optimal up to a constant factor. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theorems for ideal sheaves of zero-dimensional schemes in P2. Moreover, we give a series of examples of cuspidal curves where the family V is reducible, but where ss1(P2nC) coincides (and is abelian) for all C 2 V .
Compositio Mathematica, 1993
Handbook of Geometry and Topology of Singularities II, 2021
We report on the problem of the existence of complex and real algebraic curves in the plane with ... more 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 $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and $T$-smoothness (i.e., smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we spend special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, howev...
Trends in Mathematics, 2021
The problem we are considering came up in connection with the classification of singularities in ... more 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, where such a bound is well known, the problem is rather subtle, since the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the fibre dimension is not semicontinuous and the quasi-finite locus is not open. However, if we pass to the completed fibres in a family of rings or modules we can prove that their fibre dimension is semicontinuous under some mild conditions. We prove this in a rather general framework by introducing and using the completed and the Henselian tensor product, the proof being more involved than one might think. Finally we apply this to the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections and to the determinacy in a family of ideals.
Applicable Algebra in Engineering, Communication and Computing, 2019
We consider the actions of different groups G on the space M m,n of m × n matrices with entries i... more We consider the actions of different groups G on the space M m,n of m × n matrices with entries in the formal power series ring K[[x 1 ,. .. , x s ]], K an arbitrary field. G acts on M m,n 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 contact equivalence of functions and mappings, resp. ideals. A is called finitely G-determined if any matrix B, with entries of A − B in x 1 , ..., x s k for some k, is contained in the G-orbit of A. The purpose of this paper is to present algorithms for checking finite determinacy, to compute determinacy bounds and to compute the image T A (GA) of the tangent map to the orbit map G → GA. The tangent image is contained in the tangent space T A (GA) of the orbit GA and we apply the algorithms to prove that both spaces may be different if the field K has positive characteristic, even for contact equivalence of functions. This fact had been overlooked by several authors before. Besides this application, the algorithms of this paper may be of interest for the classification of singularities in arbitrary characteristic.
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Papers by Gert-Martin Greuel