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Liouvillian solutions of linear differential equations of order three and higher
 JOURNAL OF SYMBOLIC COMPUTATION
, 1999
"... Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closedform”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van ..."
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Cited by 32 (10 self)
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Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closedform”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semiinvariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one semiinvariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. These coefficients come “for free ” as a byproduct of our algorithm for computing semiinvariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic (1986), see also Ulmer and Weil (1996) or Fakler (1997)). In the appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm.
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.
Fast algorithms for polynomial solutions of linear differential equations
 In Proceedings of ISSAC’05
, 2005
"... Si l’on se bornait à demander les intégrales entières, le problème n’offrirait aucune difficulté. 1 Joseph Liouville, 1833. We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are ..."
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Cited by 13 (5 self)
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Si l’on se bornait à demander les intégrales entières, le problème n’offrirait aucune difficulté. 1 Joseph Liouville, 1833. We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in an integer N (that can be computed from the equation), even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(N log 3 N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O ( √ N log 2 N) bit operations. In general, the integer N is not bounded polynomially in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.
Absolute reducibility of differential operators and Galois groups
, 2004
"... A differential operator L ∈ C(x)[d/dx] is called absolutely reducible if it admits a factorization over an algebraic extension of C(x). In this paper, we give sharp bounds on the degree of the extension that is needed in order to compute an absolute factorization. Algorithms to characterize and comp ..."
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Cited by 12 (1 self)
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A differential operator L ∈ C(x)[d/dx] is called absolutely reducible if it admits a factorization over an algebraic extension of C(x). In this paper, we give sharp bounds on the degree of the extension that is needed in order to compute an absolute factorization. Algorithms to characterize and compute absolute factorizations are then elaborated. The ingredients are differential Galois theory, a grouptheoretic study of absolute factorization, and a descent technique for differential operators with coefficients in C(x).
Computing Galois groups of completely reducible differential equations
 J. Symbolic Computation
, 1998
"... We give an algorithm to calculate a presentation of the Picard–Vessiot extension associated to a completely reducible linear differential equation (i.e. an equation whose Galois group is reductive). Using this, we show how to compute the Galois group of such an equation as well as properties of the ..."
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We give an algorithm to calculate a presentation of the Picard–Vessiot extension associated to a completely reducible linear differential equation (i.e. an equation whose Galois group is reductive). Using this, we show how to compute the Galois group of such an equation as well as properties of the Galois groups of general equations. c ○ 1999 Academic Press 1.
On Symmetric Powers of Differential Operators
 Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (ISSAC’97
, 1997
"... We present alternative algorithms for computing symmetric powers of linear ordinary differential operators. Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or o ..."
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Cited by 11 (2 self)
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We present alternative algorithms for computing symmetric powers of linear ordinary differential operators. Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or over sufficiently complicated coefficient domains. The basic ideas are also applicable to other computations involving cyclic vector techniques, such as exterior powers of differential or difference operators. Introduction Let R be an integral domain of characteristic 0, D be a derivation on R, K be the quotient field of R and R[@; D] be the corresponding ring of linear differential operators with coefficients in R. Let L = P n i=0 a i @ i 2 R[@; D] with n ? 0 and an 6= 0, Y0 ; : : : ; Yn\Gamma1 be indeterminates, and consider the extension of D to K[Y0 ; : : : ; Yn\Gamma1 ] given by DY i = Y i+1 for 0 i ! n \Gamma 1 and DYn\Gamma1 = \Gammaa \Gamma1 n P n\Gamma1 i=0 a i Y i . For an...
Solving Homogeneous Linear Differential Equations of Order 4 in Terms of Equations of Lower Order
 in Terms of Equations of Smaller Order, PhD thesis, http://www.lib.ncsu.edu
, 2002
"... AXELLE CLAUDE PERSON. Solving homogeneous linear differential equations of order 4 in terms of equations of smaller order. (Under the direction of Michael F. Singer.) In this thesis we consider the problem of deciding if a fourth order linear differential equation can be solved in terms of solutions ..."
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Cited by 8 (0 self)
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AXELLE CLAUDE PERSON. Solving homogeneous linear differential equations of order 4 in terms of equations of smaller order. (Under the direction of Michael F. Singer.) In this thesis we consider the problem of deciding if a fourth order linear differential equation can be solved in terms of solutions of lower order equations. There is a group theoretic criteria which can be turned into a decision procedure for solving this problem. Once the decision has been made that a certain type of equation can be solved in terms of lower order equations we also give methods for producing the lower order equations used for solving it.
On Liouvillian Solutions of Linear Differential Equations of Order 4 and 5
 Distributed Artificial Intelligence
, 2001
"... In this paper we give the minimal possible degrees of an algebraic solution of the Riccati equation associated to an irreducible linear homogeneous differential equation of order 4 and 5, following the method used for the order 3 ([14]). We show that the algebraic degree of such a solution is bounde ..."
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Cited by 7 (0 self)
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In this paper we give the minimal possible degrees of an algebraic solution of the Riccati equation associated to an irreducible linear homogeneous differential equation of order 4 and 5, following the method used for the order 3 ([14]). We show that the algebraic degree of such a solution is bounded by 120 for the order 4 and by 55 for the order 5. With the important work done by Hessinger in [5] and the Tables computed in the sequel this leads to an algorithm to find Galois group and liouvillian solutions of equations of order 4. In the last section we construct an irreducible differential equation of degree 4 having SL(2; 7) as a Galois group and compute its liouvillian solutions. We also give the minimal polynomial over Q(x) of the algebraic solutions of the differential equation. This example illustrates a method which is easy to use in order to compute a liouvillian solution with the help of the Tables computed in this paper. 1.