Differential and difference algebra

DOI10.1007/BF01094866zbMath0666.12018MaRDI QIDQ1116979

Alexander V. Mikhalev, E. V. Pankrat'ev

Publication date: 1989

Published in: Journal of Soviet Mathematics (Search for Journal in Brave)

zbMATH Keywords

Bibliographydifferential Galois theorydifferential algebradifferential fieldsdifference algebradifferential modulesdifferential algebraic groupdifferential ringsintegration in finite form

Mathematics Subject Classification ID

Differential algebra (12H05) Other algebraic groups (geometric aspects) (14L40) Modules of differentials (13N05) Difference algebra (12H10) Research exposition (monographs, survey articles) pertaining to field theory (12-02)

Related Items (1)

Dimension polynomials of intermediate differential fields and the strength of a system of differential equations with group action

Cites Work

• A survey of Galois theory for systems of partial differential equations and its applications in physics
• Differential schemes and premodels of differential fields
• On universal derivations
• Remark on nilpotency of derivations
• Derivations of special subrings of matrix rings and regular graphs
• Lie ideals with regular and nilpotent elements and a result on derivations
• Dimension polynomials of extensions of difference fields
• Class groups and differential function fields
• On the structure presheaf of a differential ring
• Logarithmic singularities of Fuchs equations, and a criterion for the monodromy group to be finite
• A normal form and integration in finite terms for a class of elementary functions
• Permissibility of a maximally differential prime ideal
• Remarks on maximally differential prime ideals
• Power series solutions of algebraic differential equations
• On the centralizers of ideals and nil derivations
• Polynomials in denumerable indeterminates
• Derivations of PI-algebras and smoothness of noncommutative curves
• Cyclic derivation of noncommutative algebraic power series
• Inner derivations of higher orders
• Solution spaces and similarity classes of linear real meromorphic differential operator equations
• Derivations on Lie ideals in semiprime rings
• Higher \(R\)-derivations of special subrings of matrix rings
• \({\mathcal D}\)-modules et faisceaux pervers dont le support singulier est un croisement normal
• MACSYMA from F to G
• Integration in finite terms: a method for determining regular fields for the Risch algorithm
• Derivations with invertible values in rings with involution
• Derivations and Cayley derivations of generalized Cayley-Dickson algebras
• Higher derivations, normal flatness, and analytic products
• Valuations and the Ritt problem
• On solutions of differential equations which satisfy certain algebraic relations
• Nil derivations
• An algorithm for solving second order linear homogeneous differential equations
• On the calculation of some differential Galois groups
• Prolongation of integral domains
• Elementary and Liouvillian solutions of linear differential equations
• Algebraic aspects of nonlinear differential equations
• Solutions of an algebraic differential equation of the first order in a Liouvillian extension of the coefficient field
• Derivations and automorphisms of matrices symmetric relative to a canonical involution
• A cyclic derivative in noncommutative algebra
• The approximation of solutions of linear homogeneous differential equations by rational functions
• Specializations of differential kernels and the Ritt problem
• Order of Liouvillian elements satisfying an algebraic differential equation of the first order
• On universal extensions of differential fields
• Derivation towers of Lie algebras
• Minimal length of Liouville chain for solutions of an algebraic differential equation
• First order algebraic differential equations. A differential algebraic approach
• Modules of imperfection in differential algebra
• Strong normality and normality for difference fields
• The structure of strongly normal difference extensions
• Ritt schemes and torsion theory
• Lie ideals and derivations of prime rings
• Derivations of polynomial rings
• Geometry of nonlinear differential equations
• Solutions of algebraic differential equations
• On the quasi-affine scheme of a differential ring
• Derivations with nilpotent values
• On the Galois theory of difference fields
• Sur certaines algèbres de Lie de dérivations
• Derivations of higher order and commutativity of rings
• Order for systems of differential equations and a generalization of the notion of differential ring
• Typical differential dimension of the intersection of linear differential algebraic groups
• Differential extension fields of exponential type
• Differentially closed fields
• A Goldie theorem for differentiably prime rings
• Abelian extensions in Picard-Vessiot theory
• The differential rational representation algebra on a linear differential algebraic group
• On Liouville's theory of elementary functions
• Differentially Hilbertian differential fields
• Notes toward higher differential algebra
• Adjunctions and comonads in differential algebra
• Lie ideals and derivations in rings with involution
• Exterior differentiation in Lie algebras
• The model theory of differential fields revisited
• Universal derivations and universal ring constructions
• A note to general notions of the derivation and its applications
• Multisectioned partitions of integers
• Primitive derivations in free associative algebras
• Derivations and commutativity of rings
• Transcendental constants over the coefficient fields in differential elliptic function fields
• Algebraic differential equations of the first order free from parametric singularities from the differential-algebraic standpoint
• Differential filtrations and symbolic powers of regular primes
• Taylor series calculus for ring objects of line type
• Fortsetzung von Derivationen
• Differentially simple rings
• Derivations of monocomposition algebras
• Differential valuations
• Derivations of generalized quasimatrix rings
• Derivationen und Automorphismen von Algebren, in denen die Gleichung \(X^2=0\) nur trivial lösbar ist
• Constrained extensions of differential fields
• Algebraic solutions of differential equations (p-curvature and the Hodge filtration)
• On the iterates of derivations of prime rings
• Linear feedback systems and the groups of Galois and Lie
• Differential rings constructed from quasi-prime ideals
• Differentiability in local fields of prime characteristic
• On the composition of derivations
• On an extension of the Picard-Vessiot theory
• Differentiably simple rings of prime characteristic
• Liouville's theorem on functions with elementary integrals
• Integral closure of rings of solutions of linear differential equations
• Kähler differentials and differential algebra
• A notion of Krull dimension for differential rings
• On the explicit solvability of certain transcendental equations
• Differential simplicity and complete integral closure
• The inverse problem in the Galois theory of differential fields
• Ritt's question on the Wronskian
• Zero divisors in differential rings
• On the inverse problem in the Galois theory of differential fields. II
• Determination of the differentiably simple rings with a minimal ideal
• Integral closure of differential rings
• Corrections and supplements to my paper Differential modules and derivations of complete discrete valuation rings
• Some homological properties of Spencer's cohomology theory
• Canonical forms for local derivations
• Determinantal ideals, identities, and the Wronskian
• The nonminimality of the differential closure
• Exponentials in differentially algebraic extension fields
• Differential simplicity and extensions of a derivation
• Types of singularity of components of difference polynomials
• Derivations in Prime Rings
• Integrable derivations in rings of unequal characteristic
• Elementary First Integrals of Differential Equations
• On Invariant Subspaces of Locally Finite Derivations
• Modules over differential polynomial rings
• Differential Ideals in Complete Local Rings
• The Congruence Lemma for [ y p z ]
• Lie and Jordan Structure in Prime Rings with Derivations
• Derivations in Simple Rings
• On the Derivation of x n in a Ring
• Finitely Generated Extensions of Partial Difference Fields
• Sur la commutativite des anneaux differentiels
• Nilpotency of Derivatives on an Ideal
• Corrigendum to the Paper “Nilpotency of Derivations”
• Differential Formal Groups of J. F. Ritt
• Derivees logarithmiques pour une s-derivation algebrique
• Corrections to : “A conjecture in the arithmetic theory of differential equations”
• Height One Differential Ideals in Polynomial Rings
• Nilpotency of Derivations. II
• Derivations with algebraic values
• Stiff derivations of commutative rings
• Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations
• Nil Derivations and Chain Conditions in Prime Rings
• Quelques remarques a propos des s-derivations
• Derivations et automorphismes algebriques d'Anneaux premiers
• An Extension of Liouville’s Theorem on Integration in Finite Terms
• A Classification of Differential Equations of Fuchsian Class
• Some Applications of Linear Groups to Differential Equations
• Differential Identities in Prime Rings With Involution
• A Differentially Algebraic Elimination Theorem with Application to Analog Computability in the Calculus of Variations
• The nth Roots of Solutions of Linear Ordinary Differential Equations
• Algebraic Relations Among Solutions of Linear Differential Equations
• Sur la commutativite des anneaux differentiels II
• Solutions in the General Solution of Second Order Algebraic Differential Equations
• (S)-derivations algebriques sur les corps gauches et sur les anneaux premiers
• Integration in Finite Terms with Special Functions: The Logarithmic Integral
• The Risch Differential Equation Problem
• On Galois Theory of Fields with Operators
• Solution of the Inverse Problem of Differential Galois Theory in the Classical Case
• Differential Algebraic Lie Algebras
• On Second Order Linear Differential Equations with Algebraic Solutions
• A Natural Structure Theorem for Complex Fields
• A Structure Theorem for Exponential and Primitive Functions
• On the Differential Transcendence Polynomials of Finitely Generated Differential Field Extensions
• A Note on Derivations II
• DERIVATIONS AND AFFINE STRUCTURES OF SOME NILPOTENT LIE ALGEBRAS
• The Greenspan Bound for the Order of Differential Systems
• On Derivations in Prime Rings and a Question of Herstein
• Algebraic Properties of the Elementary Functions of Analysis
• On Second Order Linear Differential Equations with Algebraic Solutions on Algebraic Curves
• Differential Specializations in Integral and Algebraic Extensions
• Extensions of Difference Specializations
• Derivations and Automorphisms of Nonassociative Matrix Algebras
• Differential Algebraic Group Structures on the Plane
• Extension of derivations to modules of quotients
• Integrable derivations
• Integration in Finite Terms: The Liouville Theory
• Differential Algebra and the Simultaneous Approximation by Rational Functions of Power Series Solutions to Algebraic Functional Equations
• DIFFERENTIALLY SIMPLE RINGS WITH NO INVERTIBLE DERIVATIVES
• On the Value Group of a Differential Valuation II
• Radical irregularity of some polynomial rings
• The Derivation Algebra of a Real Division Algebra
• Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations
• Nilpotency of Derivations
• Semiprime Rings with Nilpotent Derivatives
• Extension of derivations to central simple algebras
• Derivations Into the Integral Closure
• Some properties of positive derivations on 𝑓-rings
• Summation in Finite Terms
• A Theorem on Derivations of Prime Rings with Involution
• Derivations of prime rings having power central values
• Standard and Alternative Algebras with Completely Reducible Derivation Algebras
• The derivations of M_n(C)
• Derivations on Regular Local Rings of Finitely Generated Type
• Prime model extensions for differential fields of characteristic p ≠ 0
• The Derivation Algebra of M₄⁸(C)
• Solutions of Linear Differential Equations in Function Fields of One Variable
• A generalization of WROŃSKI theorems
• The General Solution of a First Order Differential Polynomial
• Jordan Derivations on Rings
• Lie and Jordan Ideals in Prime Rings with Derivations
• On Derivations of Lie Algebras
• First Order Differential Closures of Certain Partially Ordered Fields
• A liouville theorem on integration in finite terms for line integrals^∗
• On Derivation Algebras of Malcev Algebras and Lie Triple Systems
• A Matrix Representation for Associative Algebras. II
• Differential Algebraic Subgroups of SL(2) and Strong Normality in Simple Extensions
• On Derivation Algebras of Malcev Algebras
• Derivations in central separable algebras
• An application of Ritt’s low power theorem
• A simple axiomatics for differentiation.
• On Lie algebras of differential formal groups of Ritt
• The group of automorphisms of a differential algebraic function field
• Lie Rings of Derivations of Associative Rings
• Symmetric monoidal comonads and differential algebra
• The model theory of ordered differential fields
• A Class of Differential Fields with Minimal Differential Closures
• Differential Simplicity in Regular Local Rings
• Galois groups and elementary solutions of some linear differential equations.
• Simple Lie Rings of Derivations of Commutative Rings
• Systems of n Partial Differential Equations in n Unknown Functions: the Conjecture of M. Janet
• Extensions of Differential Places
• Derivations and Automorphisms of Exterior Algebras
• The Lie Structure of a Commutative Ring with a Derivation
• On the Value Group of a Differential Valuation
• ON PICARD-VESSIOT EXTENSIONS OF DIFFERENTIAL FIELDS
• A Note on Derivations
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• The differential closure of a differential field
• A conjecture in the arithmetic theory of differential equations
• A counterexample to elimination in systems of algebraic differential equations
• On the inverse problem in difference Galois theory
• Algebraic Extensions of Difference Fields
• Pro-Algebraic Groups and the Galois Theory of Differential Fields
• PRIME IDEAL STRUCTURE IN ADDITIVE CONSERVATIVE SYSTEMS
• An Extension of Differential Galois Theory
• Complete Models of Differential Fields
• Differential Dimension Polynomials and a Fundamental Theorem on Differential Modules
• The Problem of Integration in Finite Terms
• Extending Differential Specializations
• The solution of the problem of integration in finite terms
• On Rings with a Higher Derivation
• On Spencer's Cohomology Theory for Linear Partial Differential Operators
• Radical Regularity in Differential Rings
• Homological properties of the ring of differential polynomials
• Some Applications of Differential Algebra to Ring Theory
• Shorter Notes: Extensions of Derivations
• Jacobi's Bound for the Order of Systems of First Order Differential Equations
• Infinite Nodal Noncommutative Jordan Algebras; Differentiably Simple Algebras
• The Differential Ideals [ y p z ]
• On the Inverse Problem of Galois Theory of Differential Fields
• Automorphisms and derivations of incidence algebras
• Extensions of Differential Modules Over Formal Power Series Rings
• An Eisenstein Criterion for Noncommutative Polynomials
• Differentiably simple algebras
• Jacobi's Bound for First Order Difference Equations
• Rational Functions on Differentially Closed Sets
• Integration in Finite Terms
• An Analogue of L'Hospital's Rule
• Differential Algebraic Groups
• Picard-Vessiot Theory of Linear Homogeneous Difference Equations
• Extension of derivations
• Ordered differential fields
• Higher differential algebras of discrete valuation rings
• Intégration algorithmique des fonctions élémentairement transcendantes sur une courbe algébrique

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