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Definability of linear equation systems over groups and rings

Anuj Dawar, Eryk Kopczynski, Bjarki Holm, Erich Grädel, Wied Pakusa, Arnaud Durand
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

Anuj Dawar, Erich Grädel, Bjarki Holm, Eryk Kopczynski, Wied Pakusa, Marc Herbstritt
2012 Annual Conference for Computer Science Logic
Theorem k-ideal 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

Katsunori Saito
2016 Annales de la Faculté des Sciences de Toulouse
So the domain S is not a Picard-Vessiot 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]

Katsunori Saito
2012 arXiv   pre-print
Let us consider a 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]

V. Arvind, T. C. Vijayaraghavan
2005 Lecture Notes in Computer Science
In this paper we first examine the computational complexity 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]

Phyllis J. Cassidy, Michael F. Singer
2005 arXiv   pre-print
used to classify 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)

Annette Maier
2015 Journal of Algebra
In this article, we consider the inverse Galois problem for parameterized differential 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

Lucia Di Vizio
2019 Publications mathématiques de Besançon
field (K, ∂) on the solutions 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]

Lucia Di Vizio, Charlotte Hardouin
2020 arXiv   pre-print
Notice that a 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

Lian Li, Huilin Li, Yixun Liu
1993 Annals of Pure and Applied Logic
Li 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

Carlos E. Arreche, Michael F. Singer
2017 Journal of Algebra
We apply recent results 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

Lucia Di Vizio, Charlotte Hardouin
2012 Pacific Journal of Mathematics
We show that the parametrized difference Galois 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

Michael A. Arbib, Ernest G. Manes
1980 Journal of Pure and Applied Algebra
-/is itself an 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]

Lucia DI Vizio, Charlotte Hardouin
2011 arXiv   pre-print
We show that the parameterized difference Galois 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]

Stéphanie Delaune, Steve Kremer, Daniel Pasaila
2012 Lecture Notes in Computer Science
The results strongly rely on the isomorphism between 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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