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17
Factoring and decomposing a class of linear functional systems”, Linear Algebra Appl.,
, 2008
"... This paper is dedicated to Paul A. Fuhrmann for his 70th anniversary. His scientific work, in particular Abstract Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, overdetermined, underdetermine ..."
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This paper is dedicated to Paul A. Fuhrmann for his 70th anniversary. His scientific work, in particular Abstract Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, overdetermined, underdetermined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, timedelay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M , where M (resp., M ) is a module intrinsically associated with the linear functional system Ry = 0 (resp., R z = 0). These morphisms define applications sending solutions of the system R z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a noninjective endomorphism of the module M is equivalent to the existence of a nontrivial factorization R = R 2 R 1 of the system matrix R. The corresponding system can then be integrated "in cascade". Under certain conditions, we also show that the system Ry = 0 is equivalent to a system R z = 0, where R is a blocktriangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system Ry = 0 to the integration of two independent systems R 1 y 1 = 0 and R 2 y 2 = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system Ry = 0, i.e., they allow us to compute an equivalent system R z = 0, where R is a blockdiagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given.
Linear Differential Operators for Polynomial Equations
"... Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate i ..."
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Cited by 20 (5 self)
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Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate information concerning the Galois group of P over k 0 (x) as well as over k 0 (x).
COMPUTING HOMOMORPHISMS BETWEEN HOLONOMIC DMODULES
, 2000
"... Let K ⊆ C be a subfield of the complex numbers, and let D be the ring of Klinear differential operators on R = K[x,..., xn]. If M and N are holonomic left Dmodules we present an algorithm that computes explicit generators for the finite dimensional vector space HomD(M, N). This enables us to answe ..."
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Cited by 10 (2 self)
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Let K ⊆ C be a subfield of the complex numbers, and let D be the ring of Klinear differential operators on R = K[x,..., xn]. If M and N are holonomic left Dmodules we present an algorithm that computes explicit generators for the finite dimensional vector space HomD(M, N). This enables us to answer algorithmically whether two given holonomic modules are isomorphic. More generally, our algorithm can be used to get explicit generators for Exti D (M, N) for any i.
Dmodules for Macaulay 2
"... Dmodules for Macaulay 2 is a collection of the most recent algorithms that deal with various computational aspects of the theory of Dmodules. This paper provides a brief guide, which gives examples of using the main functions of this package, as well as an overview of the core algorithms for Dmod ..."
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Cited by 6 (1 self)
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Dmodules for Macaulay 2 is a collection of the most recent algorithms that deal with various computational aspects of the theory of Dmodules. This paper provides a brief guide, which gives examples of using the main functions of this package, as well as an overview of the core algorithms for Dmodules and their applications. 1
Computing closed form solutions of integrable connections
, 2012
"... We present algorithms for computing rational and hyperexponential solutions of linear Dfinite partial differential systems written as integrable connections. We show that these types of solutions can be computed recursively by adapting existing algorithms handling ordinary linear differential syste ..."
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Cited by 4 (0 self)
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We present algorithms for computing rational and hyperexponential solutions of linear Dfinite partial differential systems written as integrable connections. We show that these types of solutions can be computed recursively by adapting existing algorithms handling ordinary linear differential systems. We provide an arithmetic complexity analysis of the algorithms that we develop. A Maple implementation is available and some examples and applications are given.
Algorithms for Algebraic Analysis
, 2000
"... One of the major goals in the field of symbolic computation of differential equations is to develop algorithms for exact or closedform solutions. This thesis studies symbolic computation of maximally overdetermined systems of linear partial differential equations by using constructions in the corre ..."
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One of the major goals in the field of symbolic computation of differential equations is to develop algorithms for exact or closedform solutions. This thesis studies symbolic computation of maximally overdetermined systems of linear partial differential equations by using constructions in the corresponding ring of differential operators vith polynomial coefficients, vhich is called the Weyl algebra D. We develop algorithms to find polynomial solutions, rational function solutions, and more generally holonomic solutions. By holonomic solutions, we mean the following: sometimes the best way to specify a function F is as the solution of a system of differential equations  this is for instance how many special functions are classically described. Our algorithm takes as input the differential equations describing F as vell as the system S that we wish to solve, and returns as output any solutions to S existing within the Dmodule generated by F. We also study aspects of the opposite problem, namely given a function F, hov can differential equations describing F be produced? We introduce the Weyl closure of an ideal I of the Weyl algebra, vhich is the set of all differential operators annihilating the common holomorphic solutions of I at a generic point. We give an algorithm to compute Weyl closure, vhich has applications to symbolic integration, and vhich ve also use to make a detailed study of ideals in the first Weyl algebra.
Weyl closure, torsion, and local cohomology of Dmodules
, 2000
"... this article, we study the following operation. ..."
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On polynomial solutions of linear partial differential and (q)difference equations
"... Abstract. We prove that the question of whether a given linear partial differential or difference equation with polynomial coefficients has nonzero polynomial solutions is algorithmically undecidable. However, for equations with constant coefficients this question can be decided very easily since s ..."
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Abstract. We prove that the question of whether a given linear partial differential or difference equation with polynomial coefficients has nonzero polynomial solutions is algorithmically undecidable. However, for equations with constant coefficients this question can be decided very easily since such an equation has a nonzero polynomial solution iff its constant term is zero. We give a simple combinatorial proof of the fact that in this case the equation has polynomial solutions of all degrees. For linear partial qdifference equations with polynomial coefficients, the question of decidability of existence of nonzero polynomial solutions remains open. Nevertheless, for such equations with constant coefficients we show that the space of polynomial solutions can be described algorithmically. We present examples which demonstrate that, in contrast with the differential and difference cases where the dimension of this space is either infinite or zero, in the qdifference case it can also be finite and nonzero. 1
Journal of Symbolic Computation 40 (2005) 979–997 Perfect bases for differential equations
, 2003
"... www.elsevier.com/locate/jsc We present a method for the construction of solutions of certain systems of partial differential equations with polynomial and power series coefficients. For this purpose we introduce the concept of perfect differential operators. Within this framework we formulate divisi ..."
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www.elsevier.com/locate/jsc We present a method for the construction of solutions of certain systems of partial differential equations with polynomial and power series coefficients. For this purpose we introduce the concept of perfect differential operators. Within this framework we formulate division theorems for polynomials and power series. They in turn yield existence theorems for solutions of systems of linear partial differential equations and algorithms to explicitly construct solutions. © 2005 Elsevier Ltd. All rights reserved.