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 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 Constructing Gröbner bases for Noetherian rings

2013 ◽  
Vol 24 (2) ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Gröbner Basis ◽  
Finite Depth ◽  
Commutative Rings ◽  
Polynomial Ideal ◽  
Constructive Proof ◽  
Noetherian Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Prime Decomposition ◽  
Separate Paper

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

scholarly journals Subalgebra Analogue to standard basis for ideal

2011 ◽  
Vol 48 (4) ◽  
pp. 458-474
Author(s):  
Junaid Khan
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Standard Basis ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis

A theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial orderings which are not necessarily well orderings [1]) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Analogue to Gröbner Basis for Ideals”, is treated in [6]. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them.

scholarly journals Finite domination and Novikov rings: Laurent polynomial rings in two variables

2015 ◽  
Vol 14 (04) ◽  
pp. 1550055
Author(s):  
Thomas Hüttemann ◽  
David Quinn
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Homotopy Equivalent ◽  
Polynomial Rings ◽  
Cochain Complex ◽  
Finitely Generated ◽  
Bounded Complex ◽  
Laurent Polynomial Ring ◽  
Free Modules

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗L R〚x, y〛[(xy)-1] and C ⊗L R[x, x-1]〚y〛[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

BIVARIATE DIMENSION POLYNOMIALS AND NEW INVARIANTS OF FINITELY GENERATED D-MODULES

2013 ◽  
Vol 23 (07) ◽  
pp. 1625-1651 ◽  
Author(s):  
CHRISTIAN DÖNCH ◽  
ALEXANDER LEVIN
Keyword(s):  
Gröbner Basis ◽  
Weyl Algebra ◽  
Isomorphism Problem ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
D Modules ◽  
Dimension Polynomial

In this paper, we generalize the classical Gröbner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely generated module over a Weyl algebra. We also present corresponding algorithms and examples of computation of such polynomials and show that bivariate dimension polynomials contain invariants of a D-module, which are not carried by its Bernstein dimension polynomial. Then we apply the obtained results to the isomorphism problem for D-modules.

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 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 A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

10.37236/8935 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Rosa Orellana ◽  
Michael Zabrocki
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Invariant Polynomials ◽  
Multivariate Polynomial ◽  
Generating Set ◽  
Combinatorial Model

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions.  We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring. 

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