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 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.

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 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.

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 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 On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

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