arXiv subjects

Christian Eder

Publications and source records attributed to Christian Eder.

At least 19 recordsLinked to original sources

arXiv2023

A Direttissimo Algorithm for Equidimensional Decomposition

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.

cs.SC↗

arXiv2022

A Signature-based Algorithm for Computing the Nondegenerate Locus of a Polynomial System

Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The nondegenerate locus of a polynomial system is the set of points where the codimension of the solution set matches the number of equations. Computing the nondegenerate locus is classically done through ideal-theoretic operations in commutative algebra such as saturation ideals or equidimensional decompositions to extract the component of maximal codimension. By exploiting the algebraic features of signature-based Gr\"obner basis algorithms we design an algorithm which computes a Gr\"obner basis of the equations describing the closure of the nondegenerate locus of a polynomial system, without computing first a Gr\"obner basis for the whole polynomial system.

cs.SC↗

arXiv2022

New efficient algorithms for computing Gr\"obner bases of saturation ideals (F4SAT) and colon ideals (Sparse-FGLM-colon)

This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical issue in non-linear algebra and more broadly in computational mathematics because of its numerous applications in engineering and computing sciences. Such applications often require geometric computing features such as representing the closure of the set difference of two solution sets to given polynomial systems. Algebraically, this boils down to computing Gr\"obner bases of colon and/or saturation polynomial ideals. In this paper, we describe and analyze new Gr\"obner bases algorithms for this task and present implementations which are more efficient by several orders of magnitude than the state-of-the-art software.

cs.SC↗

arXiv2021

msolve: A Library for Solving Polynomial Systems

We present a new open source C library \texttt{msolve} dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of \texttt{msolve} relies on Gr\''obner bases and linear algebra based algorithms for polynomial system solving. It relies on Gr\''obner basis computation w.r.t.\ the degree reverse lexicographical order, Gr\''obner conversion to a lexicographical Gr\''obner basis and real solving of univariate polynomials. We explain in detail how these three main steps of the solving process are implemented, how we exploit \texttt{AVX2} instruction processors and the more general implementation ideas we put into practice to better exploit the computational capabilities of this algorithmic framework. We compare the practical performances of \texttt{msolve} with leading computer algebra systems such as \textsc{Magma}, \textsc{Maple}, \textsc{Singular} on a wide range of systems with finitely many complex solutions, showing that \texttt{msolve} can tackle systems which were out of reach by the computer algebra software state-of-the-art.

cs.SC↗

arXiv2019

Breaking the Hidden Irreducible Polynomials Scheme

In 2019 Gómez described a new public key cryptography scheme based on ideas from multivariate public key cryptography using hidden irreducible polynomials. We show that the scheme's design has a flaw which lets an attacker recover the private key directly from the public key.

cs.CR↗

arXiv2019

Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

math.QA↗

arXiv2019

Efficient Gröbner Bases Computation over Principal Ideal Rings

In this paper we present a new efficient variant to compute strong Gröbner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$. Possibly using available factorization algorithms we may thus recursively reduce some strong Gröbner basis computations to Gröbner basis computations over fields for prime factors of $n$, at least for squarefree $n$. Considering now a computation over $R/nR$ we can run a standard Gröbner basis algorithm pretending $R/nR$ to be field. If we discover a non-invertible leading coefficient $c$, we use this information to try to split $n = ab$ with coprime $a, b$. If no such $c$ is discovered, the returned Gröbner basis is already a strong Gröbner basis for the input ideal over $R/nR$.

math.AC↗

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.

math.AC↗

arXiv2017

Modular Techniques For Noncommutative Gröbner Bases

In this note, we extend modular techniques for computing Gröbner bases from the commutative setting to the vast class of noncommutative $G$-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Gröbner basis can only be expected to generate the given ideal, with high probability. We have implemented our algorithm in the computer algebra system {\sc{Singular}} and give timings to compare its performance with that of other instances of Buchberger's algorithm, testing examples from $D$-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.

math.RA↗

arXiv2016

New Strategies for Standard Bases over Z

Experiences with the implementation of strong Gröbner bases respectively standard bases for polynomial rings over principal ideal rings are explained: different strategies for creating the pair set, methods to avoid coefficient growth and a normal form algorithm for non-global orderings.

math.AC↗

arXiv2016

GBLA -- Gröbner Basis Linear Algebra Package

This is a system paper about a new GPLv2 open source C library GBLA implementing and improving the idea of Faugère and Lachartre (GB reduction). We further exploit underlying structures in matrices generated during Gröbner basis computations in algorithms like F4 or F5 taking advantage of block patterns by using a special data structure called multilines. Moreover, we discuss a new order of operations for the reduction process. In various different experimental results we show that GBLA performs better than GB reduction or Magma in sequential computations (up to 40% faster) and scales much better than GB reduction for a higher number of cores: On 32 cores we reach a scaling of up to 26. GBLA is up to 7 times faster than GB reduction. Further, we compare different parallel schedulers GBLA can be used with. We also developed a new advanced storage format that exploits the fact that our matrices are coming from Gröbner basis computations, shrinking storage by a factor of up to 4. A huge database of our matrices is freely available with GBLA.

cs.SC↗

arXiv2014

A survey on signature-based Gröbner basis computations

This paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.

math.AC↗

arXiv2014

Predicting zero reductions in Gröbner basis computations

Since Buchberger's initial algorithm for computing Gröbner bases in 1965 many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of Gebauer and Möller. A relatively new approach are signature-based criteria which were first used in Faugère's F5 algorithm in 2002. For regular input sequences these criteria are known to compute no zero reduction at all. In this paper we give a detailed discussion on zero reductions and the corresponding syzygies. We explain how the different methods to predict them compare to each other and show advantages and drawbacks in theory and practice. With this a new insight into algebraic structures underlying Gröbner bases and their computations might be achieved.

math.AC↗

arXiv2013

An analysis of inhomogeneous signature-based Gröbner basis computations

In this paper we give an insight into the behaviour of signature-based Gröbner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the performance. The lost connection between polynomial degree and signature degree can disallow lots of reductions and can lead to an overhead in the computations. On the other hand, the way critical pairs are sorted and corresponding s-polynomials are handled in signature-based algorithms is a very efficient one, strongly connected to sorting w.r.t. the well-known sugar degree of polynomials.

math.AC↗

arXiv2012

Improving incremental signature-based Groebner basis algorithms

In this paper we describe a combination of ideas to improve incremental signature-based Groebner basis algorithms having a big impact on their performance. Besides explaining how to combine already known optimizations to achieve more efficient algorithms, we compare the quite different effects on the two best-known algorithms in this area, F5 and G2V, both from a theoretical and a practical point of view.

math.AC↗

arXiv2011

Signature-based algorithms to compute Groebner bases

This paper describes a Buchberger-style algorithm to compute a Groebner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.

math.AC↗

arXiv2011

F5C: a variant of Faugere's F5 algorithm with reduced Groebner bases

Faugere's F5 algorithm computes a Groebner basis incrementally, by computing a sequence of (non-reduced) Groebner bases. The authors describe a variant of F5, called F5C, that replaces each intermediate Groebner basis with its reduced Groebner basis. As a result, F5C considers fewer polynomials and performs substantially fewer polynomial reductions, so that it terminates more quickly. We also provide a generalization of Faugere's characterization theorem for Groebner bases.

math.AC↗

arXiv2010

Modifying Faugère's F5 Algorithm to ensure termination

The structure of the F5 algorithm to compute Gröbner bases makes it very efficient. However, while it is believed to terminate for so-called regular sequences, it is not clear whether it terminates for all inputs. This paper has two major parts. In the first part, we describe in detail the difficulties related to a proof of termination. In the second part, we explore three variants that ensure termination. Two of these have appeared previously only in dissertations, and ensure termination by checking for a Gröbner basis using traditional criteria. The third variant, F5+, identifies a degree bound using a distinction between "necessary" and "redundant" critical pairs that follows from the analysis in the first part. Experimental evidence suggests this third approach is the most efficient of the three.

math.AC↗