A survey on signature-based Gröbner basis computations

Eder, Christian; Faugère, Jean-Charles

doi:10.1145/2815111.2815156

Cited by 9 publications

(8 citation statements)

References 46 publications

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“…Gröbner bases started as a generalization of Gauss elimination to polynomials. They have since come back to their roots in linear algebra by the advent of F4 and F5 type algorithms which try to arrange computations so that sparse linear algebra can exploited [7]. Our method draws on linear algebra in bases of monomials too, and is inspired by these developments in computer algebra.…”

Section: Introductionmentioning

confidence: 99%

Detecting binomiality

Conradi

Kahle

2015

Advances in Applied Mathematics

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Abstract. Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and source of examples is the simplification of steady state equations of chemical reaction networks. For homogeneous ideals we give an efficient, Gröbner-free algorithm for binomiality detection, based on linear algebra only. On inhomogeneous input the algorithm can only give a sufficient condition for binomiality. As a remedy we construct a heuristic toolbox that can lead to simplifications even if the given ideal is not binomial.

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Section: Introductionmentioning

confidence: 99%

Detecting binomiality

Conradi

Kahle

2015

Advances in Applied Mathematics

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“…In particular from an involutive basis, many invariants of the ideal like its Hilbert function can be easily read off and, in principle, one even knows a basis of the first syzygy module (Seiler, 2009b). Thus ideas like a Hilbert-driven Buchberger algorithm (Traverso, 1996) or exploiting syzygies for the detection of reductions to zero (Möller et al, 1992) (see more generally (Eder and Faugère, 2017) for a recent survey on signature based algorithms) can significantly increase the efficiency. Binaei et al (2016) report on some preliminary results in particular concerning the first point.…”

Section: Implementations and Experimentsmentioning

confidence: 99%

Deterministic genericity for polynomial ideals

Hashemi

Schweinfurter

Seiler

2018

Journal of Symbolic Computation

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We consider several notions of genericity appearing in algebraic geometry and commutative algebra. Special emphasis is put on various stability notions which are defined in a combinatorial manner and for which a number of equivalent algebraic characterisations are provided. It is shown that in characteristic zero the corresponding generic positions can be obtained with a simple deterministic algorithm. In positive characteristic, only adapted stable positions are reachable except for quasi-stability which is obtainable in any characteristic.

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“…The main reference concerning the Matrix-F5 algorithm are Bardet's PhD thesis [Bar04], Bardet, Faugère and Salvy's analysis of the complexity of the F5 algorithm in [BFS14] and Eder and Faugère's survey of F5 algorithms in [EF14]. We first recall some basic facts about matrix-algorithm to compute Gröbner bases, and present the Matrix-F5 algorithm.…”

Section: Matrix-f5mentioning

confidence: 99%

Matrix-F5 algorithms over finite-precision complete discrete valuation fields

Vaccon

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Let (f 1 , . . . , f s ) ∈ Q p [X 1 , . . . , X n ] s be a sequence of homogeneous polynomials with p-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since Q p is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gröbner basis with respect to a monomial order w. We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals f 1 , . . . , f i are weakly-w-ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic.Two variants of that strategy are available, depending on whether one lean more on precision or time-complexity. For the analysis of these algorithms, we study the loss of precision of the Gauss row-echelon algorithm, and apply it to an adapted Matrix-F5 algorithm. Numerical examples are provided.Moreover, the fact that under such hypotheses, Gröbner bases can be computed stably has many applications. Firstly, the mapping sending (f 1 , . . . , f s ) to the reduced Gröbner basis of the ideal they span is differentiable, and its differential can be given explicitly. Secondly, these hypotheses allows to perform lifting on the Grobner bases, from Z/pFinally, asking for the same hypotheses on the highest-degree homogeneous components of the entry polynomials allows to extend our strategy to the affine case.

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A survey on signature-based Gröbner basis computations

Cited by 9 publications

References 46 publications

Detecting binomiality

Detecting binomiality

Deterministic genericity for polynomial ideals

Matrix-F5 algorithms over finite-precision complete discrete valuation fields

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