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On Gröbner Basis Detection for Zero-dimensional Ideals
[article]

2011
*
arXiv
*
pre-print

The Gr\"obner

arXiv:1105.0433v1
fatcat:odcrss2xnndv7gnas7nryiegve
*basis**detection*(GBD) is defined as follows: Given a set of polynomials, decide whether there exists -and if "yes" find- a term order such that the set of polynomials is a Gr\"obner*basis*... This problem was shown to be NP-hard by showing the NP-completeness of a variant of GBD called '*Structural**Gröbner**basis**detection*' (SGBD). ... The*Gröbner**basis**detection*zero-dimensional ideals is defined as follows. ...##
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Middle-Solving Grobner bases algorithm for cryptanalysis over finite fields
[article]

2015
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arXiv
*
pre-print

Finally, by

arXiv:1507.03480v1
fatcat:fnod5dt55beg7h2aj4j3u2vwp4
*detecting*the temporary*basis*during the computation of*Grobner*bases and then extracting the univariate polynomials contained unique solution in the temporary*basis*, a heuristic strategy named ... Secondly, we give and prove the degree bound of the polynomials appeared during the computation of*Grobner**basis*after field polynomials is added. ... So we use Middle-Solving strategy to*detect**Gröbner**basis*for 1 ,, ff i G , a*Gröbner**basis*for 1 ,, i ff , where [2,..., ] im . ...##
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Detecting binomiality

2015
*
Advances in Applied Mathematics
*

For homogeneous ideals we give an efficient, Gr\"obner-free algorithm for binomiality

doi:10.1016/j.aam.2015.08.004
fatcat:3seg2pa6yvbmxamzf63rqusgai
*detection*, based on linear algebra only. ...*Gröbner**basis*computations can be reduced to the homogeneous case by an easy trick.*Detection*of binomiality can not (Example 4.1). ... The*Gröbner**basis*consists of 169 elements each of it with huge rational functions as coefficients. The*structure*that we observed above is completely lost. ...##
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An algorithm for computing the universal Gröbner Basis of graph ideals
[article]

2019
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arXiv
*
pre-print

The universal Gr\"obner

arXiv:1307.7224v3
fatcat:n7gnj4p7ofgozlzi3o5twj2hki
*basis*of an ideal is a Gr\"obner*basis*with respect to all term orders simultaneously. ... set and on a recent, efficiently computable algorithmic characterization of the Graver*basis*of the ideal. ... The*structure*of the universal*Gröbner**basis*of a toric ideal of a graph G, was characterized theoretically in [10, Theorem 3.4.] . ...##
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Predicting zero reductions in Gröbner basis computations
[article]

2014
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arXiv
*
pre-print

With this a new insight into algebraic

arXiv:1404.0161v1
fatcat:yxjjwgtwinhlxfucnoiujlbx4i
*structures*underlying Gr\"obner bases and their computations might be achieved. ... Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965 many attempts have been taken to*detect*zero reductions in advance. ... Due to the fact that i is the reduced*Gröbner**basis*for 〈 f 1 , . . . , f i 〉 we have more*structure*to exploit. ...##
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On Gröbner Bases in the Context of Satisfiability-Modulo-Theories Solving over the Real Numbers
[chapter]

2013
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Lecture Notes in Computer Science
*

polynomials from the input set and (2) it can compute an inconsistent subset of the input constraints if the

doi:10.1007/978-3-642-40663-8_18
fatcat:suwdi3g4nnbldbwzf2cc6wuq5e
*Gröbner**basis*contains 1. ... We modify Buchberger's algorithm by implementing a new update operator to optimize the*Gröbner**basis*and provide two methods to handle inequalities. ... Definition 1 (*Gröbner**basis*) Let P ⊆ R \ {0}. A finite set G ⊆ P is called a*Gröbner**basis*(GB) of P if {lt(g) | g ∈ G} = {lt(p) | p ∈ P } . Let lc(p) = 1 for all p ∈ G. ...##
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Complexity of Gröbner basis detection and border basis detection

2012
*
Theoretical Computer Science
*

*Gröbner*

*basis*

*detection*(GBD) is defined as follows: given a set of polynomials, decide whether there exists -and if "yes" find -a term order such that the set of polynomials is a

*Gröbner*

*basis*. ... In this work, we define the border

*basis*

*detection*problem and study its computational complexity. More specifically, we prove that the border

*basis*

*detection*is NP-complete. Organisation. ... For this, they introduced a related problem called SGBD (

*Structural*

*Gröbner*

*basis*

*detection*) which was shown to be NP-complete by a reduction from the set packing problem. ...

##
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Improving incremental signature-based Groebner basis algorithms
[article]

2012
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arXiv
*
pre-print

In this paper we describe a combination of ideas to improve incremental signature-based Groebner

arXiv:1201.6472v3
fatcat:kr53be4pwnexhgqa55fdfkcrde
*basis*algorithms having a big impact on their performance. ... In the end, we only need two out of these three elements for a*Gröbner**basis*; in a minimal*Gröbner**basis*we would discard poly(f ). ... G i+1 (since B i is already a reduced*Gröbner**basis*). ...##
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Signature-Based Gröbner Basis Algorithms --- Extended MMM Algorithm for computing Gröbner bases
[article]

2013
*
arXiv
*
pre-print

By this view, this paper aims to give an easier way to understand signature-based Gr\"obner

arXiv:1308.2371v1
fatcat:fjo5cgfstrgajcjeljyyhegfva
*basis*algorithms. ...*Detecting*redundant computations/critical pairs. ...*Gröbner**basis*for f 1 , · · · , f m . ...##
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Residual Generator Design for Non-Linear, Polynomial Systems - A Gröbner Basis Approach

2000
*
IFAC Proceedings Volumes
*

The example also shows how a fault

doi:10.1016/s1474-6670(17)37485-2
fatcat:fbzd63oifzgztjiy5zgc4y7gia
*detectability*/isolability analysis can be made during the design. ... The design procedure is applied in a simulation study on a non-linear system, where it is showed how multiplicative and additive faults are*detected*and isolated. ... A design variable, apart from choosing h i polynomials in (4) , is the variable ordering when calculating the*Gröbner**basis*. Different variable orderings highly influences the resulting*basis*. ...##
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Groebner bases of reaction networks with intermediate species
[article]

2018
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arXiv
*
pre-print

For standard networks, this extra condition can be visually explored from the network

arXiv:1804.01381v1
fatcat:viu3jlz7krck3gx2dh3dsbtble
*structure*alone. ... We show that a Groebner*basis*of the steady state ideal of the core network always lifts to a Groebner*basis*of the steady state ideal of the extended network by means of linear algebra, with a suitable ... Let G be a*Gröbner**basis*of I. By Theorem 3.4, G is a*Gröbner**basis*of I. ...##
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An analysis of inhomogeneous signature-based Gröbner basis computations
[article]

2013
*
arXiv
*
pre-print

In this paper we give an insight into the behaviour of signature-based Gr\"obner

arXiv:1203.6186v4
fatcat:o677ranuavbuzbqoctp4foahsy
*basis*algorithms, like F5, G2V or SB, for inhomogeneous input. ... Whenever one wants to compute a*Gröbner**basis*for an inhomogeneous ideal with F5 the idea of homogenization can be used: 3. Cut down G h to a*Gröbner**basis*G for F . ... Let us have a look at a quite generic signature-based*Gröbner**basis*algorithm. ...##
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Improved Computation of Involutive Bases
[article]

2017
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arXiv
*
pre-print

algorithm, we modify the algorithm, given by Seiler, to compute a linear change of coordinates for a given homogeneous ideal so that the new ideal (after performing this change) possesses a finite Pommaret

arXiv:1705.03441v1
fatcat:zk2xrs37ajbrtl7l5s4eddcwfm
*basis*... Schreyer in his master thesis proposed a slight modification of Buchberger's algorithm to compute a*Gröbner**basis*for the module of syzygies of a*Gröbner**basis*. ... For a*Gröbner**basis*G = {g 1 , . . . , g s } the set {s ij | 1 ≤ i < j ≤ s} forms a*Gröbner**basis*for Syz(g 1 , . . . , g s ) w.r.t. ≺ s . Example 13. Let F = {xy − x, x 2 − y} ⊂ [x, y]. ...##
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Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge
[article]

2005
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arXiv
*
pre-print

The

arXiv:math/0508287v1
fatcat:p3cgmfkjyna6dixj43e5723wca
*Gröbner**basis*in Proposition 3.5 is rarely minimal. ... An easy modification to algorithm (3.1) may*detect*if I is not saturated. ...##
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A new incremental algorithm for computing Groebner bases

2010
*
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10
*

At a typical step, one is given a

doi:10.1145/1837934.1837944
dblp:conf/issac/GaoGV10
fatcat:ssgzhnq2kbedxhjhocucqr3p5e
*Gröbner**basis*G for an ideal I and any polynomial g, and it is desired to compute a*Gröbner**basis*for the new ideal I, g , obtained from I by joining g. ... Our algorithm computes*Gröbner*bases for I, g and (I : g) simultaneously. ... Our algorithm will use a reduced*Gröbner**basis*G as in F5C, but the crucial difference is that we introduce a so-called "super topreduction" to*detect*"useless" polynomials. ...
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