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On Solutions Of Linear Ordinary Differential Equations In Their Coefficient Field
 Journal of Symbolic Computation
, 1991
"... this paper, we consider the following specific subproblem in this area: given a differential field k, g 2 k, and a linear ordinary differential operator L with coefficients in k, can we decide in a finite number of steps whether L(y) = g has a solution in k, and in the affirmative, can we find one ( ..."
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Cited by 44 (6 self)
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this paper, we consider the following specific subproblem in this area: given a differential field k, g 2 k, and a linear ordinary differential operator L with coefficients in k, can we decide in a finite number of steps whether L(y) = g has a solution in k, and in the affirmative, can we find one (or all) such solution(s)?
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 20 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
Liouvillian solutions of third order differential equations
 SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
"... The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper ..."
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Cited by 7 (0 self)
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The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for 4 finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations.
Linear Differential Equations In Exponential Extensions
"... We present an algorithm to compute rational solutions of linear dierential equations with coecients in exponential extensions of monomial extensions of a base eld. We focus on the system of generators describing the extension and show why some of the generators sets are more \suitable" tha ..."
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Cited by 1 (0 self)
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We present an algorithm to compute rational solutions of linear dierential equations with coecients in exponential extensions of monomial extensions of a base eld. We focus on the system of generators describing the extension and show why some of the generators sets are more \suitable" than others. This results partially improves and generalizes the method presented in [Sin91] to nd liouvillian solutions of linear differential equations with coecients in liouvillian extension of C(x).
Solving Linear Systems of Differential and Difference Equations with Respect to a Part of the Unknowns
, 2005
"... Abstract—The existence of solutions y to a linear differential or difference system whose components y, …, y belong to a given class of functions and the problem of constructing such components are i1 im considered. DOI: 10.1134/S0965542506020059 ..."
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Abstract—The existence of solutions y to a linear differential or difference system whose components y, …, y belong to a given class of functions and the problem of constructing such components are i1 im considered. DOI: 10.1134/S0965542506020059