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Gerhard Pfister

Publications and source records attributed to Gerhard Pfister.

At least 19 recordsLinked to original sources
arXiv2025

Grauert's Approximation Theorem in any Characteristic and Applications

In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of "nested" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers. The purpose of this article is to extend Grauert's division and approximation theorem to convergent power series over arbitrary real valued fields of any characteristic. As an application, which was actually the motivation for this article, we derive the existence of a convergent semiuniversal deformation for an isolated singularity and a splitting lemma for not necessarily isolated hypersurface singularities over any real valued field.

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arXiv2025

The Splitting Lemma in any Characteristic

We give a simple proof of the splitting lemma in singularity theory, also known as generalized Morse lemma, for formal power series over arbitrary fields. Our proof for the uniqueness of the residual part in any characteristic is new and was previously unknown in characteristic two. Beyond the formal case, we give proofs for algebraic power series and for convergent real and complex analytic power series, which are new for non-isolated singularities.

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arXiv2024

Classification of simple 0-dimensional isolated complete intersection singularities

The aim of this article is the classification of simple 0-dimensional isolated complete intersection singularities in positive characteristic. As usual, a singularity is called simple or 0-modal if there are only finitely many isomorphism classes of singularities into which the given singularity can deform. The notion of simpleness was introduced by V. I. Arnold and the classification of low modality singularities has become a fundamental task in singularity theory. Simple complex analytic isolated complete intersection singularities (ICIS) were classified by M. Giusti. However, the classification in positive characteristic requires different methods and is much more involved. The final result is nevertheless similar to the classification in characteristic 0 with some additional normal forms in low characteristic. The theoretical results in this paper mainly concern families of ICIS that are formal in the fiber and algebraic in the base (formal deformation theory is not sufficient). In particular, we give a definition of modality in this situation and prove its semicontinuity.

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arXiv2024

Moduli Parameters of Complex Singularities with Non-Degenerate Newton Boundary

Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is unbounded with respect to modality and Milnor number. This method is based on proving that all right-equivalence classes within a mu-constant stratum can be represented by a single normal form derived from a regular basis of a suitably selected special fiber. While both Arnold's and our preceding work on normal forms addresses the determination of a normal form family containing the given germ, this paper takes the next natural step: We present an algorithm for computing for a given germ the values of the moduli parameters in its normal form family, that is, a normal form equation in its stable equivalence class. This algorithm will be crucial for understanding the moduli stacks of such singularities. The implementation of this algorithm, along with the foundational classification techniques, is implemented in the library arnold.lib for the computer algebra system Singular.

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arXiv2023

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

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 goal of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes are not applicable in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with SINGULAR. Our methods are characteristic-independent and provide a different proof for the classification in characteristic 0, showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and the modality, which we prove.

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arXiv2021

Using Semicontinuity for Standard Bases Computations

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. The most important special cases are perhaps when A is the ring of integers resp. when A is a polynomial ring over some field in finitely many parameters. Given I as an ideal in the polynomial ring over A, we compute first a standard basis modulo a prime number p, resp. by specializing the parameter to a constant. We then use the "highest corner" of the specialized ideal to cut off high order terms from the polynomials during the standard basis computation over K to get the speed up. An important fact is that we can choose p as an arbitrary prime resp. as an arbitrary constant, not just a "lucky" resp. "random" one. Correctness of the algorithm will be deduced from a general semicontinuity theorem due to the first two authors. The computer algebra system Singular provides already the functionality to realize the algorithm and we present several examples illustrating its power.

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arXiv2021

The Delta Invariant and Simultaneous Normalization for Families of Isolated Non-Normal Singularities

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 that it behaves upper semicontinuous in flat families parametrized by an arbitrary principal ideal domain. Moreover, if the fibers contain no isolated points, then the familly admits a fiberwise normalization iff the delta invariant is locally constant. The results generalize results by Teissier and Chiang-Hsieh--Lipman for families of reduced curve singularities and provide possible improvements for algorithms to compute the genus of a curve.

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arXiv2021

Computing integral bases via localization and Hensel lifting

We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and completion, together with the Chinese remainder theorem, to reduce the problem to the task of finding integral bases for the branches of each singularity of the curve. To solve the latter task, in turn, we work with suitably truncated Puiseux expansions. In contrast to van Hoeij's algorithm, which also relies on Puiseux expansions (but pursues a different strategy), we use Hensel's lemma as a key ingredient. This allows us at some steps of the algorithm to compute factors corresponding to conjugacy classes of Puiseux expansions, without actually computing the individual expansions. In this way, we make substantially less use of the Newton-Puiseux algorithm. In addition, our algorithm is inherently parallel. As a result, it outperforms in most cases any other algorithm known to us by far. Typical applications are the computation of adjoint ideals and, based on this, the computation of Riemann-Roch spaces and the parametrization of rational curves.

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arXiv2021

On Delta for parameterized Curve Singularities

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.

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arXiv2020

Classification of Complex Singularities with Non-Degenerate Newton Boundary

In his groundbreaking work on classification of singularities with regard to right and stable equivalence of germs, Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with either modality less than or equal to two or Milnor number less than or equal to 16. Moreover, he has described an algorithmic classifier, which determines the type of a given such singularity. In the present paper, we extend Arnold's work to a large class of singularities which is unbounded with regard to modality and Milnor number. We develop an algorithmic classifier, which determines a normal form for any singularity with corank less than or equal to two which is equivalent to a germ with non-degenerate Newton boundary in the sense of Kouchnirenko. In order to realize the classifier, we prove a normal form theorem: Suppose K is a mu-constant stratum of the jet space which contains a germ with a non-degenerate Newton boundary. We first observe that all germs in K are equivalent to some germ with the same fixed non-degenerate Newton boundary. We then prove that all right-equivalence classes of germs in K can be covered by a single normal form obtained from a regular basis of an appropriately chosen special fiber. All algorithms are implemented in the library arnold.lib for the computer algebra system Singular.

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arXiv2020

Semicontinuity of Singularity Invariants in Families of Formal Power Series

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 fibers in a family of modules we can prove that their fiber 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 as 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.

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arXiv2019

On the Primary Decomposition of Some Determinantal Hyperedge Ideal

In this paper we describe the method which we applied to successfully compute the primary decomposition of a certain ideal coming from applications in combinatorial algebra and algebraic statistics regarding conditional independence statements with hidden variables. While our method is based on the algorithm for primary decomposition by Gianni, Trager and Zacharias, we were not able to decompose the ideal using the standard form of that algorithm, nor by any other method known to us.

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arXiv2018

Standard Bases over Euclidean Domains

In this paper we state and explain techniques useful for the computation of strong Gröbner and standard bases over Euclidean domains: First we investigate several strategies for creating the pair set using an idea by Lichtblau. Then we explain methods for avoiding coefficient growth using syzygies. We give an in-depth discussion on normal form computation resp. a generalized reduction process with many optimizations to further avoid large coefficients. These are combined with methods to reach GCD-polynomials at an earlier stage of the computation. Based on various examples we show that our new implementation in the computer algebra system Singular is, in general, more efficient than other known implementations.

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arXiv2017

Unimodular ICIS, a Classifier

We present the algorithms for computing the normal form of unimodular complete intersection surface singularities classified by C. T. C. Wall. He indicated in the list only the $μ$-constant strata and not the complete classification in each case. We give a complete list of surface unimodular singularities. We also give the description of a classifier which is implemented in computer algebra system \singular.

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arXiv2017

An uniform General Neron Desingularization in dimension one

We give an uniform General Neron Desingularization for one dimensional local rings with respect to morphisms which coincide modulo a high power of the maximal ideal. The result has interesting applications in the case of Cohen-Macaulay rings.

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arXiv2017

Bad Primes in Computational Algebraic Geometry

Computations over the rational numbers often suffer from intermediate coefficient swell. One solution to this problem is to apply the given algorithm modulo a number of primes and then lift the modular results to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. In many applications, however, there is no efficient way of excluding bad primes. In this note, we describe a technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the selected set of primes is large enough. We give a number of illustrating examples which are implemented using the computer algebra system Singular and the programming language Julia. We discuss applications of our technique in computational algebraic geometry.

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