scholarly journals Divisors on graphs, Connected flags, and Syzygies

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Farbod Shokrieh
Keyword(s):  
Gröbner Basis ◽  
Betti Numbers ◽  
Point Of View ◽  
Grobner Basis ◽  
Generating Set ◽  
Term Order ◽  
Syzygy Module ◽  
International Audience ◽  
The Given ◽  
Minimal Generating Set

International audience We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.

Determining of Level Sets for a Fuzzy Surface Using Gröbner Basis

2015 ◽  
Vol 4 (2) ◽  
pp. 1-14
Author(s):  
Hamed Farahani ◽  
Sajjad Rahmany ◽  
Abdolali Basiri
Keyword(s):  
Level Sets ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Statistical Nature ◽  
Numerical Examples ◽  
Incomplete Datasets ◽  
The Given

In this paper, a manner to determine the level sets of a fuzzy surface using the benefits of Gröbner basis is presented. Fuzzy surfaces are constructed from incomplete datasets or from data that contain uncertainty which has not statistical nature. The authors firstly define the concept of level sets for the fuzzy surfaces. Then, employing Gröbner bases benefits a criterion is proposed for when the level sets of the fuzzy surface are nonempty sets. Moreover, a new algorithm is designed to determine the level sets. The big advantage of the proposed method lies in the fact that it attains all members of the level sets of the fuzzy surface at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach.

scholarly journals Intersection of Ideals in a Polynomial Ring over a Dual Valuation Domain

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Regis F. Babindamana ◽  
Andre S. E. Mialebama Bouesso
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Valuation Domain ◽  
Grobner Basis ◽  
Generating Set

Let V be a valuation domain and let A=V+εV be a dual valuation domain. We propose a method for computing a strong Gröbner basis in R=A[x1,…,xn]; given polynomials f1,…,fs∈R, a method for computing a generating set for Syz(f1,…,fs)={(h1,…,hs)∈Rs∣h1f1+⋯+hsfs=0} is given; and, finally, given two ideals I=〈f1,…,fs〉 and J=〈g1,…,gr〉 of R, we propose an algorithm for computing a generating set for I∩J.

scholarly journals Algebraic Cryptanalysis Scheme of AES-256 Using Gröbner Basis

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Kaixin Zhao ◽  
Jie Cui ◽  
Zhiqiang Xie
Keyword(s):  
Gröbner Basis ◽  
Variable Order ◽  
Polynomial Equations ◽  
Construction Process ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Algebraic Cryptanalysis ◽  
Term Order ◽  
Basis Construction ◽  
Depth Study

The zero-dimensional Gröbner basis construction is a crucial step in Gröbner basis cryptanalysis on AES-256. In this paper, after performing an in-depth study on the linear transformation and the system of multivariate polynomial equations of AES-256, the zero-dimensional Gröbner basis construction method is proposed by choosing suitable term order and variable order. After giving a detailed construction process of the zero-dimensional Gröbner basis, the necessary theoretical proof is presented. Based on this, an algebraic cryptanalysis scheme of AES-256 using Gröbner basis is proposed. Analysis shows that the complexity of our scheme is lower than that of the exhaustive attack.

scholarly journals Strongly Robust Toric Ideals in Codimension 2

2019 ◽  
Vol 10 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Seth Sullivant
Keyword(s):  
Positive Answer ◽  
Homogeneous Ideal ◽  
Toric Ideals ◽  
Grobner Basis ◽  
Toric Ideal ◽  
Generating Set ◽  
Graver Basis ◽  
Minimal Generating Set ◽  
Gale Diagrams ◽  
Universal Gröbner Basis

A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set.  For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials.  We characterize the codimension 2  strongly robust toric ideals by their Gale diagrams.  This give a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.

scholarly journals The Discrete Fundamental Group of the Associahedron

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Christopher Severs ◽  
Jacob White
Keyword(s):  
Fundamental Group ◽  
Lattice Theory ◽  
Homotopy Theory ◽  
Equivalence Classes ◽  
Point Of View ◽  
Cluster Algebras ◽  
Type B ◽  
Simple Description ◽  
Generating Set ◽  
International Audience

International audience The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \par L'associahèdre est un objet bien etudié que l'on retrouve dans plusieurs contextes. Par exemple, il est associé à la théorie des opérades, à l'étude des partitions non-croisées, à la théorie des treillis et plus récemment aux algèbres dámas. Nous étudions cet objet par le biais de la théorie des homotopies discretes. En bref cette théorie signifie qu'un cycle de longueur 5 (sur le squelette de l'associahèdre) est considéré comme étant le bord d'un trou combinatoire, alors qu'un cycle de longueur 4 peut être contracté sans problème. Les classes d'homotopies discrètes sont donc des classes d'équivalence de cycles de longueurs 5. Nous donnons une description simple de ces classes d'équivalence et identifions un ensemble de générateurs du groupe correspondant (abélien) d'homotopies discrètes. Nous d'ecrivons également les liens entre notre construction et les algèbres d'amas.

Multi-graded Betti numbers of path ideals of trees

2017 ◽  
Vol 16 (01) ◽  
pp. 1750018 ◽  
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Explicit Description ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Generating Set ◽  
Graded Betti Numbers ◽  
Minimal Free Resolutions ◽  
Vertex Sets ◽  
Minimal Generating Set

A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for [Formula: see text]-ary trees.

Gröbner bases for syzygy modules of border bases

2014 ◽  
Vol 13 (06) ◽  
pp. 1450003 ◽  
Author(s):  
Martin Kreuzer ◽  
Markus Kriegl
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Polynomial Ideal ◽  
Grobner Bases ◽  
Order Ideal ◽  
Grobner Basis ◽  
Syzygy Module ◽  
Syzygy Modules ◽  
Term Ordering ◽  
Algorithmic Form

Given an order ideal 𝒪 and an 𝒪-border basis of a 0-dimensional polynomial ideal, it was shown by Huibregtse that the liftings of the neighbor syzygies (i.e. of the fundamental syzygies of neighboring border terms) form a system of generators for the syzygy module of the border basis. We elaborate on Huibregtse's proof and transform it into explicit algorithmic form. Based on this, we are able to exhibit explicit conditions on a module term ordering τ such that the liftings of the neighbor syzygies are in fact a τ-Gröbner basis. Finally, we construct term orderings satisfying these conditions in an explicit algorithmic way.

scholarly journals Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Gröbner Basis

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Nicolas Thiéry
Keyword(s):  
Gröbner Basis ◽  
The Other ◽  
Permutation Groups ◽  
Grobner Basis ◽  
Generating Sets ◽  
Invariant Rings ◽  
Invariant Ring ◽  
International Audience ◽  
System Of Parameters ◽  
Special Case

International audience We present a characteristic-free algorithm for computing minimal generating sets of invariant rings of permutation groups. We circumvent the main weaknesses of the usual approaches (using classical Gröbner basis inside the full polynomial ring, or pure linear algebra inside the invariant ring) by relying on the theory of SAGBI- Gröbner basis. This theory takes, in this special case, a strongly combinatorial flavor, which makes it particularly effective. Our algorithm does not require the computation of a Hironaka decomposition, nor even the computation of a system of parameters, and could be parallelized. Our implementation, as part of the library $permuvar$ for $mupad$, is in many cases much more efficient than the other existing software.

scholarly journals A Deformation of the Orlik-Solomon Algebra

2016 ◽  
Vol 118 (2) ◽  
pp. 183
Author(s):  
István Heckenberger ◽  
Volkmar Welker
Keyword(s):  
Commutative Ring ◽  
Associative Algebra ◽  
Gröbner Basis ◽  
Hilbert Series ◽  
Fiber Type ◽  
Free Associative Algebra ◽  
Grobner Basis ◽  
The Given

A deformation of the Orlik-Solomon algebra of a matroid $\mathfrak{M}$ is defined as a quotient of the free associative algebra over a commutative ring $R$ with $1$. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as $R$-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.

scholarly journals Binomial Edge Ideals with Quadratic Gröbner Bases

10.37236/698 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Marilena Crupi ◽  
Giancarlo Rinaldo
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideal ◽  
Grobner Basis ◽  
Edge Ideals ◽  
Term Order ◽  
Vertex Set

We prove that a binomial edge ideal of a graph $G$ has a quadratic Gröbner basis with respect to some term order if and only if the graph $G$ is closed with respect to a given labelling of the vertices. We also state some criteria for the closedness of a graph $G$ that do not depend on the labelling of its vertex set.

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