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 Gröbner bases and products of coefficient rings

2002 ◽  
Vol 65 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Graham H. Norton ◽  
Ana Sӑlӑgean
Keyword(s):  
Gröbner Basis ◽  
Chinese Remainder Theorem ◽  
Gröbner Bases ◽  
Commutative Rings ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Know How ◽  
Elementary Method ◽  
Coefficient Ring ◽  
Finite Direct Product

Suppose that A is a finite direct product of commutative rings. We show from first principles that a Gröbner basis for an ideal of A[x1,…,xn] can be easily obtained by ‘joining’ Gröbner bases of the projected ideals with coefficients in the factors of A (which can themselves be obtained in parallel). Similarly for strong Gröbner bases. This gives an elementary method of constructing a (strong) Gröbner basis when the Chinese Remainder Theorem applies to the coefficient ring and we know how to compute (strong) Gröbner bases in each factor.

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 An extension of Buchberger’s criteria for Gröbner basis decision

2010 ◽  
Vol 13 ◽  
pp. 111-129
Author(s):  
John Perry
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Characterization Theorem ◽  
Polynomial Systems ◽  
Polynomial Ideal ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Case Scenario ◽  
Worst Case ◽  
New Criterion

AbstractTwo fundamental questions in the theory of Gröbner bases are decision (‘Is a basisGof a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known thatGis a Gröbner basis if and only if a certain set of polynomials (theS-polynomials) satisfy a certain property. In general there arem(m−1)/2 of these, wheremis the number of polynomials inG, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systemsGfor which the new criterion has dramatic impact, reducing the worst-case scenario fromm(m−1)/2 S-polynomials tom−1.

scholarly journals Characteristic polynomials of finitely generated modules over Weyl algebras

2000 ◽  
Vol 61 (3) ◽  
pp. 387-403 ◽  
Author(s):  
Alexander B. Levin
Keyword(s):  
Characteristic Polynomial ◽  
Gröbner Basis ◽  
Weyl Algebra ◽  
Characteristic Polynomials ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Weyl Algebras ◽  
Dimension Polynomial ◽  
Finitely Generated Modules

In this paper we modify the classical Gröbner basis technique and prove the existence of a characteristic polynomial in two variables associated with a finitely generated module over a Weyl algebra. We determine invariants of such a polynomial and show that some of the invariants are not carried by the Bernstein dimension polynomial of the module.

scholarly journals Note on Non-Commutative Semi-Local Rings

1960 ◽  
Vol 17 ◽  
pp. 161-166 ◽  
Author(s):  
Yukitoshi Hinohara
Keyword(s):  
Local Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Noetherian Rings ◽  
Local Rings ◽  
Finitely Generated

Our aim in this note is to generalize some topological results of commutative noetherian rings to non-commutative rings. As a supplemental remark of [2] we prove in § 1 that any right ideal of a complete right semi-local ring is closed, and thatfor any finitely generated right module M over a complete right semi-local ring Λ where J is the Jacobson radical of Λ.

The hypergeometric system for one-loop triangle integral

2019 ◽  
Vol 34 (35) ◽  
pp. 1950232
Author(s):  
Xiu-Yi Yang ◽  
Hong-Na Li
Keyword(s):  
Parameter Space ◽  
Gröbner Basis ◽  
Hypergeometric Series ◽  
Polynomial Ideal ◽  
Convergence Theory ◽  
Grobner Basis ◽  
Convergence Region ◽  
Multiple Series

We derive the holonomic hypergeometric system for the [Formula: see text] function with two equal virtual masses, and present the expression of [Formula: see text] in hypergeometric series in corresponding convergent region. Combining the Horn’s convergence theory with Gröbner basis of polynomial ideal, one can calculate the convergence region of the corresponding multiple series concretely. Using the system given here, one can analytically continue [Formula: see text] to whole parameter space.

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.

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