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.

A Gap Theorem for Differentially Algebraic Power Series

Number Theory ◽  
1991 ◽  
pp. 211-214
Author(s):  
Leonard Lipshitz ◽  
Lee A. Rubel
Keyword(s):  
Power Series ◽  
Gap Theorem ◽  
Algebraic Power Series

scholarly journals ON DIVISION OF QUASIANALYTIC FUNCTION GERMS

2013 ◽  
Vol 24 (13) ◽  
pp. 1350111 ◽  
Author(s):  
KRZYSZTOF JAN NOWAK
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Local Ring ◽  
Taylor Series ◽  
Necessary Condition ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power ◽  
Sufficient And Necessary Condition

We establish the following criterion for divisibility in the local ring [Formula: see text] of those quasianalytic function germs at 0 ∈ ℝn which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two function germs in [Formula: see text] is that of their Taylor series at 0 ∈ ℝn in the formal power series ring.

Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic

2013 ◽  
Vol 56 (4) ◽  
pp. 673-683
Author(s):  
K. Ayadi ◽  
M. Hbaib ◽  
F. Mahjoub
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Diophantine Approximation ◽  
Positive Characteristic ◽  
Rational Approximations ◽  
Positive Degree ◽  
Formal Power ◽  
Algebraic Power Series

Abstract.In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the typewhere (A, B, C) ∊ (𝔽q[X])2 × 𝔽*q [X]. In particular, under some conditions on the polynomials A, B and C, we will give well approximated elements satisfying this equation.

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 Continuous homomorphisms and rings of injective dimension one

2012 ◽  
Vol 110 (2) ◽  
pp. 181
Author(s):  
Shou-Te Chang ◽  
I-Chiau Huang
Keyword(s):  
Power Series ◽  
Injective Dimension ◽  
One Dimensional ◽  
Injective Modules ◽  
Valuation Rings ◽  
Dimension One ◽  
Injective Hulls ◽  
Essential Extensions ◽  
Formal Power ◽  
The Ideal

Let $S$ be an $R$-algebra and $\mathfrak a$ be an ideal of $S$. We define the continuous hom functor from $R$-mod to $S$-mod with respect to the $\mathfrak a$-adic topology on $S$. We show that the continuous hom functor preserves injective modules iff the ideal-adic property and ideal-continuity property are satisfied for $S$ and $\mathfrak a$. Furthermore, if $S$ is $\mathfrak a$-finite over $R$, we show that the continuous hom functor also preserves essential extensions. Hence, the continuous hom functor can be used to construct injective modules and injective hulls over $S$ using what we know about $R$. Using the continuous hom functor we can characterize rings of injective dimension one using symmetry for a special class of formal power series subrings. In the Noetherian case, this enables us to construct one-dimensional local Gorenstein domains. In the non-Noetherian case, we can apply the continuous hom functor to a generalized form of the $D+M$ construction. We may construct a class of domains of injective dimension one and a series of almost maximal valuation rings of any complete DVR.

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.

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