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

Prabhanjan Ananth, Ambedkar Dukkipati
2011 arXiv   pre-print
The Gr\"obner 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.  ...

### Middle-Solving Grobner bases algorithm for cryptanalysis over finite fields [article]

Wansu Bao, Heliang Huang
2015 arXiv   pre-print
Finally, by 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  .  ...

### Detecting binomiality

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

### An algorithm for computing the universal Gröbner Basis of graph ideals [article]

Yannis C. Stamatiou, Christos Tatakis
2019 arXiv   pre-print
The universal Gr\"obner 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.] .  ...

### Predicting zero reductions in Gröbner basis computations [article]

Christian Eder
2014 arXiv   pre-print
With this a new insight into algebraic 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.  ...

### On Gröbner Bases in the Context of Satisfiability-Modulo-Theories Solving over the Real Numbers [chapter]

Sebastian Junges, Ulrich Loup, Florian Corzilius, Erika Ábrahám
2013 Lecture Notes in Computer Science
polynomials from the input set and (2) it can compute an inconsistent subset of the input constraints if the 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.  ...

### Complexity of Gröbner basis detection and border basis detection

Prabhanjan V. Ananth, Ambedkar Dukkipati
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.  ...

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

Christian Eder
2012 arXiv   pre-print
In this paper we describe a combination of ideas to improve incremental signature-based Groebner 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).  ...

### Signature-Based Gröbner Basis Algorithms --- Extended MMM Algorithm for computing Gröbner bases [article]

Yao Sun
2013 arXiv   pre-print
By this view, this paper aims to give an easier way to understand signature-based Gr\"obner basis algorithms.  ...  Detecting redundant computations/critical pairs.  ...  Gröbner basis for f 1 , · · · , f m .  ...

### Residual Generator Design for Non-Linear, Polynomial Systems - A Gröbner Basis Approach

Erik Frisk
2000 IFAC Proceedings Volumes
The example also shows how a fault 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.  ...

### Groebner bases of reaction networks with intermediate species [article]

2018 arXiv   pre-print
For standard networks, this extra condition can be visually explored from the network 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.  ...

### An analysis of inhomogeneous signature-based Gröbner basis computations [article]

Christian Eder
2013 arXiv   pre-print
In this paper we give an insight into the behaviour of signature-based Gr\"obner 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.  ...

### Improved Computation of Involutive Bases [article]

Bentolhoda Binaei and Amir Hashemi and Werner M. Seiler
2017 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 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].  ...

### Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge [article]

Niels Lauritzen
2005 arXiv   pre-print
The Gröbner basis in Proposition 3.5 is rarely minimal.  ...  An easy modification to algorithm (3.1) may detect if I is not saturated.  ...

### A new incremental algorithm for computing Groebner bases

Shuhong Gao, Yinhua Guan, Frank Volny
2010 Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10
At a typical step, one is given a 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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