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Definability of linear equation systems over groups and rings

2013
*
Logical Methods in Computer Science
*

*equation*

*systems*

*over*finite

*groups*

*and*

*rings*from the viewpoint

*of*logical (inter-)

*definability*. ... Our results indicate that all solvability problems for

*linear*

*equation*

*systems*that separate fixed-point logic with counting from PTIME can be reduced to solvability

*over*commutative

*rings*. ...

*Systems*

*of*

*linear*

*equations*. We consider

*systems*

*of*

*linear*

*equations*

*over*

*groups*

*and*

*rings*whose

*equations*

*and*variables are indexed by arbitrary sets, not necessarily ordered. ...

##
###
Definability of linear equation systems over groups and rings

2012
*
Annual Conference for Computer Science Logic
*

Theorem k-ideal

doi:10.4230/lipics.csl.2012.213
dblp:conf/csl/DawarGHKP12
fatcat:trwv7mjlurbcndckxjlrlodrhm
*rings*FP-red.⇒ cyclic*groups**of*prime power order.TheoremEvery FO+slv F -formula is equivalent to a formula*of*the form slv(x,ȳ). ϕ M , 1 , with ϕ M quanti er-free.Theorem k-ideal*rings*... slv(x,ȳ). ϕ M (x,ȳ), 1 , with ϕ M quanti er-free.Proof illustration: (nesting*of*solvability) slv(r,s). slv(x,ȳ). ϕ(r,s,x,ȳ),Outer*system*: S Inner*system*: I[r,s] Proof illustration: (nesting*of*solvability ...##
###
On the definition of the Galois group of linear differential equations

2016
*
Annales de la Faculté des Sciences de Toulouse
*

So the domain S is not a Picard-Vessiot

doi:10.5802/afst.1521
fatcat:jxvb52ltvjcchns3mnymaia5i4
*ring*but a generated by a fundamental*system**of*solutions*of**linear*differential*equations**over*K. ... The*ring*C(x)[exp x, (exp x) −1 ] is a Picard-Vessiot*ring**over*C(x) for a*linear*differential*equation*Y = Y with the fundamental*system**of*solution Z = exp x. ...##
###
On the definition of the Galois group of linear differential equations
[article]

2012
*
arXiv
*
pre-print

Let us consider a

arXiv:1212.4070v1
fatcat:m6up2srowzhhbkqtrkajsuvkny
*linear*differential*equation**over*a differential field K. ... For a differential field extension L/K generated by a fundamental*system**of*the*equation*, we show that Galois*group*according to the general Galois theory*of*Umemura coincides with the Picard-Vessiot Galois ... We will compare the Galois*group**of*generated*over*k by a*system**of*solutions*of*the*linear*differential*equation*Y ′ = AY, A ∈ M n (K), (13)*and*the Galois*group**of*Picard-Vessiot extension for the*equation*...##
###
The Complexity of Solving Linear Equations over a Finite Ring
[chapter]

2005
*
Lecture Notes in Computer Science
*

In this paper we first examine the computational complexity

doi:10.1007/978-3-540-31856-9_39
fatcat:iqesp3pmsbhltk22sdouahgqyq
*of*the problem LCON*defined*as follows: given a matrix A*and*a column vector b*over*Z, determine if Ax = b is a feasible*system**of**linear**equations*... We prove the same upper bound results for the problem*of*testing feasibility*of*Ax = b*over*finite*rings*R with unity, where R is given as part*of*the input as a table. ... For discussions*and*comments on this work,*and*for bringing to our notice the problems with using a notion*of*rank to test the feasibility*of**linear**equations*modulo composites, we are very grateful to ...##
###
Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
[article]

2005
*
arXiv
*
pre-print

used to classify

arXiv:math/0502396v1
fatcat:ea5335nbfbao3jjcblvlww43ae
*systems**of*second order*linear*differential*equations*. ... We present a Galois theory*of*parameterized*linear*differential*equations*where the Galois*groups*are*linear*differential algebraic*groups*, that is,*groups**of*matrices whose entries are functions*of*the ... One can proceed in an analogous fashion with integrable*systems**of**linear*differential*equations**and**define*a Galois*group*that is a collection*of*transformations*of*solutions*of*a*linear*differential ...##
###
On the parameterized differential inverse Galois problem over k((t))(x)

2015
*
Journal of Algebra
*

In this article, we consider the inverse Galois problem for parameterized differential

doi:10.1016/j.jalgebra.2014.12.039
fatcat:6kwoq7deqremplqfcmqzmfecje
*equations**over*k((t))(x) with k any field*of*characteristic zero*and*use the method*of*patching*over*fields due to ... As an application, we prove that every connected semisimple k((t))-split*linear*algebraic*group*is a parameterized Galois*group**over*k((t))(x). ... I would like to thank Julia Hartmann*and*Michael Wibmer for helpful discussions during the preparation*of*this manuscript. ...##
###
Action of an endomorphism on (the solutions of) a linear differential equation

2019
*
Publications mathématiques de Besançon
*

field (K, ∂) on the solutions

doi:10.5802/pmb.28
fatcat:w2wweuoctncl5lm2d6vnekm7za
*of*a*linear*differential*equation**defined**over*(K, ∂). ... -The purpose*of*this survey is to provide the reader with a user friendly introduction to the two articles [8]*and*[9] , which give a Galoisian description*of*the action*of*an endomorphism*of*a differential ...*of*a differential field (K, ∂) on the solutions*of*a*linear*differential*equation**defined**over*(K, ∂). ...##
###
Galois theories for q-difference equations: comparison theorems
[article]

2020
*
arXiv
*
pre-print

Notice that a

arXiv:1205.1696v3
fatcat:pbrrc4oj55e5zjo3mxfkgerevu
*linear*$q$-difference*equation*with meromorphic coefficients always admits a basis*of*meromorphic solutions, as proven by Praagman. ...*of*meromorphic solutions*of*such*equations*. ... We would like to thanks the referee for her or his attentive reading*and*the useful remarks. ...##
###
A decision algorithm for linear sentences on a PFM

1993
*
Annals of Pure and Applied Logic
*

Li

doi:10.1016/0168-0072(93)90097-w
fatcat:acbwkcgzzfelrhpxpvcq2ozbri
*and*Y. Liu, A decision algorithm for*linear*sentences on a PFM, Annals*of*Pure*and*Applied Logic 59 (1993) 273-286. ... Acknowledgements We would like to thank Professor Zhao Ying for his helpful discussions,*and*the referee for many useful suggestions. ... Application*and*conclusion A finitely generated Abelian*group*is a finitely generated module*over*the*ring**of*integers. ...##
###
Galois groups for integrable and projectively integrable linear difference equations

2017
*
Journal of Algebra
*

We apply recent results

doi:10.1016/j.jalgebra.2017.02.032
fatcat:oe4m5dyo45cf7ka2wgs4mriq6u
*of*Sch\"{a}fke*and*Singer to characterize which*groups*can occur as Galois*groups**of*integrable or projectively integrable*linear*difference*systems*. ... We consider first-order*linear*difference*systems**over*$\mathbb{C}(x)$, with respect to a difference operator $\sigma$ that is either a shift $\sigma:x\mapsto x+1$, $q$-dilation $\sigma:x\mapsto qx$ with ... The*group*G*of*σ-k-algebra automorphisms*of*this*ring*is called the σ-Galois*group**and*is a*linear*algebraic*group**defined**over*k σ . ...##
###
Descent for differential Galois theory of difference equations: confluence andq-dependence

2012
*
Pacific Journal of Mathematics
*

We show that the parametrized difference Galois

doi:10.2140/pjm.2012.256.79
fatcat:yteeseswufgufkabziu4nr6zga
*group*(with respect to a convenient derivation*defined*in the text)*of*the Jacobi Theta function can be considered as the Galoisian counterpart*of*the heat ...*equation*. ... Wibmer*and*the anonymous referee for their attentive reading*of*the manuscript*and*their comments. We are particularly indebted to M. Wibmer for the proof Proposition 1.16. ...##
###
Machines in a category

1980
*
Journal of Pure and Applied Algebra
*

-/is itself an

doi:10.1016/0022-4049(80)90090-0
fatcat:eylerllivff7holujcxxug6j74
*equationally*-*definable*class (add the elements*of*X as nullary operations*and*then add the*equations*that express the*linearity**of*G.) ...*Systems**over*a*ring*are*linear**systems*as in 1.2 but replacing R with an arbitrary*ring*R. ... Hankel realization theorem for adjoint*systems*. A bisequence*of*x'-morphisms ofform Hk .Llachines in a caleyor? ...##
###
Descent for differential Galois theory of difference equations. Confluence and q-dependency
[article]

2011
*
arXiv
*
pre-print

We show that the parameterized difference Galois

arXiv:1103.5067v2
fatcat:lzsc6oxr5rfvnmxwc2s2ei4fue
*group*(with respect to a convenient derivation*defined*in the text)*of*the Jacobi Theta function can be considered as the Galoisian counterpart*of*the heat ...*equation*. ... This*group*is*defined**over**of*C E*and*measures all differential relations satisfied by the solutions*of*the qdifference*equation*with respect to δ x*and*δ q . ...##
###
Security Protocols, Constraint Systems, and Group Theories
[chapter]

2012
*
Lecture Notes in Computer Science
*

The results strongly rely on the isomorphism between

doi:10.1007/978-3-642-31365-3_15
fatcat:hwxd5cuwejhgzgiqqs6e5h32ze
*group*theories*and**rings*. This allows us to reduce the problem under study to the problem*of*solving*systems**of**equations**over**rings*. ... We provide several new decidability*and*complexity results, notably for*equational*theories which have applications in security protocols, such as exclusive or*and*Abelian*groups*which may additionally ... leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n • 258865, project ProSecure,*and*...
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