scholarly journals The Ideal Membership Problem in Non-Commutative Polynomial Rings

1996 ◽  
Vol 22 (1) ◽  
pp. 27-48 ◽  
Author(s):  
F.LEON PRITCHARD
Keyword(s):  
Polynomial Rings ◽  
Membership Problem ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

scholarly journals Gröbner–Shirshov Bases for Replicated Algebras

2017 ◽  
Vol 24 (04) ◽  
pp. 563-576 ◽  
Author(s):  
P.S. Kolesnikov
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Membership Problem ◽  
Leibniz Algebras ◽  
Universal Approach ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

scholarly journals The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

2021 ◽  
Vol 17 (4) ◽  
pp. 1-29
Author(s):  
Monaldo Mastrolilli
Keyword(s):  
Sufficient Conditions ◽  
Constraint Satisfaction Problems ◽  
Algorithmic Problem ◽  
Membership Problem ◽  
Efficient Computation ◽  
Constrained Problems ◽  
Constraint Language ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if f ϵ I . This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the IMP for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer’s dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems (CSPs) over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body Semi-Definite Programming (SDP) relaxations, identifying therefore the borderline of tractability for constraint language problems. This article is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares (SoS) proofs [O’Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the IMP tractability for combinatorial ideals implies bounded coefficients in SoS proofs.

scholarly journals Local zero estimates and effective division in rings of algebraic power series

2018 ◽  
Vol 2018 (737) ◽  
pp. 111-160 ◽  
Author(s):  
Guillaume Rond
Keyword(s):  
Power Series ◽  
Necessary Condition ◽  
Effective Solution ◽  
Gap Theorem ◽  
Division Theorem ◽  
Ideal Membership ◽  
Formal Power ◽  
Algebraic Power Series ◽  
Ideal Membership Problem ◽  
The Ideal

AbstractWe give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

Border bases and kernels of homomorphisms and of derivations

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Janusz Zieliński
Keyword(s):  
Polynomial Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
The Ideal

AbstractBorder bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.

EXTENSIONS OF α-REFLEXIVE RINGS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250013 ◽  
Author(s):  
Liang Zhao ◽  
Xiaosheng Zhu
Keyword(s):  
Polynomial Rings ◽  
Rigid Ring ◽  
Basic Properties ◽  
Armendariz Ring ◽  
The Ideal

We introduce the notion of an α-reflexive ring to extend the concept of a reflexive ring and that of an α-rigid ring. We first consider some basic properties of α-reflexive rings, including some examples needed in the process. We prove that a ring R is α-rigid if and only if R is a reduced α-reflexive ring with α a monomorphism. We next investigate the α-reflexivity of some kinds of polynomial rings. It is shown that if R is a reduced α-reflexive ring with α(1) = 1, then R[x]/(xn) is an α-reflexive ring, where (xn) is the ideal generated by xn. Moreover, for an Armendariz ring R, we prove that R is α-reflexive if and only if R[x] is α-reflexive if and only if R[x; x-1] is α-reflexive. As a sequence, some known results relating to reflexive rings are obtained as corollaries of these results.

Idempotent Ideals and Noetherian Polynomial Rings

1982 ◽  
Vol 25 (1) ◽  
pp. 48-53 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Power Series ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Power Series Rings ◽  
Descending Chain Condition ◽  
The Ideal

AbstractIf R is a commutative Noetherian ring and I is a nonzero ideal of R, it is known that R+I[x] is a Noetherian ring exactly when I is idempotent, and so, when R is a domain, I = R and R has identity. In this paper, the noncommutative analogues of these results, and the corresponding ones for power series rings, are proved. In the general case, the ideal I must satisfy the idempotent condition that TI = T for each right ideal T of R contained in I. It is also shown that when every ideal of R satisfies this condition, and when R satisfies the descending chain condition on right annihilators, R must be a finite direct sum of simple rings with identity.

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