scholarly journals Grobner Basis Over Ideals of Multivariate Polynomial Ring: قواعد جروبنر فوق مثاليَّات الحدوديَّات بأكثر من متغير

2019 ◽  
Vol 3 (2) ◽  
Author(s):  
Khaled Suleiman Al-Akla
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Main Axis ◽  
Structural Approach ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Multiple Variables ◽  
Definition Of

Grobner basis are considered one of the modern mathematical tools which has become of interest for the researchers in all fields of mathematics. Grobner basis are generally polynomials with multiple variables that has certain characteristics. it's includes two main axis:                                                                            1- The first axis we have presented the definition of Grobner basis and their properties. 2- The second axis we have studied some applications of Grobner basis, and we give some examples about its. The goal of these paper is to identify Grobner basis and some algorithms related to how to find them and talked about the most important applications, including: the issue of belonging and the issue of containment, and to reach our goal to follow the analytical and structural approach, we defined these basis and we have many results, The Grosvenor we obtained is not alone in general and to be single, some additional conditions must be set on these basis, and we conclude that Grobner basis have many applications in the solutions of algebraic equations in more than one transformer and in many fields.

scholarly journals An Algorithm to Compute the H-Bases for Ideals of Subalgebras

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Rabia ◽  
Muhammad Ahsan Binyamin ◽  
Nazia Jabeen ◽  
Adnan Aslam ◽  
Kraidi Anoh Yannick
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Important Ingredient ◽  
Multivariate Polynomial

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.

scholarly journals GRÖBNER BASIS METHODS FOR STRUCTURING AND ANALYSING COMPLEX INDUSTRIAL EXPERIMENTS

2000 ◽  
Vol 07 (04) ◽  
pp. 285-300 ◽  
Author(s):  
GIOVANNI PISTONE ◽  
EVA RICCOMAGNO ◽  
HENRY P. WYNN
Keyword(s):  
Linear Model ◽  
Polynomial Ring ◽  
Research Program ◽  
Gröbner Basis ◽  
Algebraic Structures ◽  
Grobner Basis ◽  
Industrial Experiments ◽  
Physical Experiments

This work extends the research program of the authors into the design and analysis of complex experiments. It shows how the special algebraic structures studied in the polynomial ring algebra and Gröbner basis environment can be exploited for situations in which there is blocking, nesting, crossing and so on, or where groups of factors are "favoured" over others. The connection is made between the Gröbner basis methods and the more classical symbolic formalism associated, for example, with Generalised Linear Model packages, such as S-Plus and Glim. Examples are given from physical experiments in engine mapping.

Methods to solve algebraic equations in cryptanalysis

2010 ◽  
Vol 45 (1) ◽  
pp. 107-136 ◽  
Author(s):  
Igor Semaev ◽  
Michal Mikuš
Keyword(s):  
Gröbner Basis ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Sat Solving

ABSTRACT The goal of the present paper is a survey of methods to solve equation systems common in cryptanalysis. The methods depend on the equation representation and fall into three categories: Gröbner basis algorithms, SAT-solving methods and Agreeing-Gluing algorithms.

IDEAL COSET INVARIANTS FOR SURFACE-LINKS IN ℝ4

2013 ◽  
Vol 22 (09) ◽  
pp. 1350052 ◽  
Author(s):  
YEWON JOUNG ◽  
JIEON KIM ◽  
SANG YOUL LEE
Keyword(s):  
Normal Form ◽  
Polynomial Ring ◽  
Gröbner Basis ◽  
Quotient Ring ◽  
Grobner Basis ◽  
Link Invariant ◽  
Link Invariants ◽  
Classical Link ◽  
Marked Graph

In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.

scholarly journals A note on multivariate polynomial division and Gröbner bases

2015 ◽  
Vol 97 (111) ◽  
pp. 43-48
Author(s):  
Aleksandar Lipkovski ◽  
Samira Zeada
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Heuristic Approach ◽  
Grobner Bases ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Leading Term ◽  
Monomial Ordering

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

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.

On Finite Noncommutative Gröbner Bases

2020 ◽  
Vol 27 (03) ◽  
pp. 381-388
Author(s):  
Yatma Diop ◽  
Djiby Sow
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Ideal Formula ◽  
Monomial Ordering ◽  
Noncommutative Polynomial

It is well known that in the noncommutative polynomial ring in serveral variables Buchberger’s algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].

scholarly journals Symbolic-Numerical Modeling of the Influence of Damping Moments on Satellite Dynamics

2018 ◽  
Vol 173 ◽  
pp. 05008
Author(s):  
Sergey A. Gutnik ◽  
Vasily A. Sarychev
Keyword(s):  
Numerical Modeling ◽  
Asymptotic Stability ◽  
Gröbner Basis ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Real Roots ◽  
Spatial Oscillations ◽  
Damping Torque ◽  
Orbital Coordinate System

The dynamics of a satellite on a circular orbit under the influence of gravitational and active damping torques, which are proportional to the projections of the angular velocity of the satellite, is investigated. Computer algebra Gröbner basis methods for the determination of all equilibrium orientations of the satellite in the orbital coordinate system with given damping torque and given principal central moments of inertia were used. The conditions of the equilibria existence depending on three damping parameters were obtained from the analysis of the real roots of the algebraic equations spanned by the constructed Gröbner basis. Conditions of asymptotic stability of the satellite equilibria and the transition decay processes of the spatial oscillations of the satellite at different damping parameters have also been obtained.

Sign in / Sign up

Export Citation Format

Share Document