Linear systems over localizations of rings

This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization, and present an algorithm for the computation of the symbolic power of a given ideal in a commutative ring. We also provide algorithms to compute local closures for certain non-commutative rings with respect to Ore sets with enough commutativity.

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“…In the commutative setting it is an important ingredient for solving linear systems over commutative localizations (Posur (2018)).…”

Section: Algorithmic Toolboxmentioning

confidence: 99%

Constructive arithmetics in Ore localizations enjoying enough commutativity

Hoffmann

Levandovskyy

2021

Journal of Symbolic Computation

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“…Here, a complete solver means that we can compute all the solutions to the system. This problem has been studied in the case of coherent rings [19], which includes the case of the polynomial ring K[ ]. Here we describe the approach that is the actual implementation provided in the Sage package dd_functions.…”

Section: Linear Systems On Localized Ringsmentioning

confidence: 99%

Simple Differentially Definable Functions

Jiménez-Pastor

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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D-finite functions satisfy linear differential equations with polynomial coefficients. The solutions to this type of equations may have singularities determined by the zeros of their leading coefficient. There are algorithms to desingularize the equations, i.e., remove singularities from the equation that do not appear in its solutions. However, classical computations of closure properties (such as addition, multiplication, etc.) with D-finite functions return equations with extra zeros in the leading coefficient. In this paper we present theory and algorithms based on linear algebra to control the leading coefficients when computing these closure properties and we also extend this theory to the more general class of differentially definable functions. CCS CONCEPTS• Mathematics of computing → Mathematical software; Generating functions; Ordinary differential equations.

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“…If we retain those witnesses and pass them to other algorithms which need them as input, we may save costly calls of oracles. An example of a computer algebra problem 3 where this philosophy was successfully applied can be found in [28]. 3.…”

Section: Remark 21 (Bishop's Constructive Mathematics For Practitionmentioning

confidence: 99%

A Constructive Approach to Freyd Categories

Posur

2020

Appl Categor Struct

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We discuss Peter Freyd’s universal way of equipping an additive category $$\mathbf {P}$$ P with cokernels from a constructive point of view. The so-called Freyd category $$\mathcal {A}(\mathbf {P})$$ A ( P ) is abelian if and only if $$\mathbf {P}$$ P has weak kernels. Moreover, $$\mathcal {A}(\mathbf {P})$$ A ( P ) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems $$X \cdot \alpha = \beta $$ X · α = β for morphisms $$\alpha $$ α and $$\beta $$ β in $$\mathbf {P}$$ P . We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for $$\mathbf {P}$$ P that helps solving linear systems in $$\mathbf {P}$$ P and even in the iterated Freyd category construction $$\mathcal {A}( \mathcal {A}(\mathbf {P})^{\mathrm {op}} )$$ A ( A ( P ) op ) , which can be identified with the category of finitely presented covariant functors on $$\mathcal {A}(\mathbf {P})$$ A ( P ) . The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.

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Linear systems over localizations of rings

Cited by 6 publications

References 9 publications

Constructive arithmetics in Ore localizations enjoying enough commutativity

Constructive arithmetics in Ore localizations enjoying enough commutativity

Simple Differentially Definable Functions

A Constructive Approach to Freyd Categories

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