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...
Linear Differential Operators for Polynomial Equations (2008)
Olivier Cormier, Michael F. Singer, Barry M. Trager, Barry M, Felix Ulmer, ...
Given a squarefree polynomial P k 0 [x, y], k 0 a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P = 0 (when P is...
Olivier Cormier, Michael F. Singer, Felix Ulmer, ...
In this paper we show how to compute the geometric Galois group G Q(x) of a polynomial f 2 Q(x)[Y] by considering the associated linear differential equation L f (Y) = 0 (and constructions of it)...
Differential Equations and Finite Groups (2007)
this paper is to construct a linear differential equation L (either in matrix form or in scalar form) over the differential field k := Q(z), with derivation
Olivier Cormier, Campus De Beaulieu, Michael F. Singer, Felix Ulmer
maths.univ-rennes1.fr In this paper we show how to compute the Galois group G of a polynomial f 2 Q(x)[Y] by factoring the associated linear differential equation Lf (Y) = 0 (and constructions of it)...
Linear Dierential Operators for Polynomial Equations (2007)
Olivier Cormier, Felix Ulmer, Michael F. Singer, Barry M. Trager
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...
Linear differential operators for polynomial equations (2002)
Olivier Cormier, Michael F. Singer, Barry M. Trager, Felix Ulmer
Given a squarefree polynomial P ∈ k0[x, y], k0 a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P = 0 (when P is...
Liouvillian solutions of linear differential equations of order three and higher (1999)
Mark Van Hoeij, Felix Ulmer, Jacques-arthur Weil
Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closed-form”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made...
How to Solve Linear Differential Equations - an outline (1999)
There are several definitions of closed form solutions of linear differential equations. In this paper we look for the so called Liouvillian solutions. Through examples, we give an overview of how...
Liouvillian Solutions of Linear Differential Equations of Order Three and Higher (1998)
Mark Van Hoeij, Jean-Francois Ragot, Felix Ulmer, Jacques-Arthur Weil
this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semi-invariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one...
Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations (1998)
Michael F. Singer, Felix Ulmer
this paper we show that the index of a 1-reducible subgroup of the differential Galois
Differential Equations and Finite Groups (1998)
this paper is to construct a linear differential equation L (either in matrix form or in scalar form) over the differential field k := Q(z), with derivation
Constructing a Third Order Linear Differential Equation (1997)
In this paper, using the approach of Hurwitz and the necessary conditions given in [4,6], we construct a third order linear differential equation whose differential Galois group is the primitive...
Galois Groups of Second and Third Order Linear Differential Equations (1997)
Michael F. Singer, Felix Ulmer
this paper we show how factorization properties of these symmetric powers can be used to determine structural properties of the galois groups of second and third order linear differential equation....
Linear Differential Equations and Products of Linear Forms (1997)
Michael F. Singer, Felix Ulmer
We show that liouvillian solutions of an n-th order linear differential equation L(y) = 0 are related to semi-invariant forms of the differential Galois group of L(y) = 0 which factor into linear...
Liouvillian Solutions of Linear Differential Equations of Order Three and Higher (1997)
Mark Van Hoeij, Jean-François Ragot, Felix Ulmer, Jacques-Arthur Weil
In [SUl97], Singer and Ulmer gave an algorithm to compute Liouvillian ("closed-form") solutions of homogeneous linear differential equations. However, there were several efficiency problems...
Irreducible Linear Differential Equations of Prime Order (1995)
this paper we consider linear differential equations of the form
Note on Kovacic's algorithm (1995)
Felix Ulmer, Jacques-Arthur Weil
There exists algorithms to find Liouvillian solutions of second order homogeneous linear differential equations (see [7, 17]). In this paper, we show how, by carefully combining the techniques of...
Linear Differential Equations and Liouvillian Solutions (1994)
eld which is an extension of k, and \Delta be the derivation on K (resp ffi on k). We say K is a differential field extension of k if \Delta and ffi coincide on k. 1.2. Now, we say that a solution of...
Necessary Conditions for Liouvillian Solutions of (Third Order) Linear Differential Equations (1993)
Michael F. Singer, Felix Ulmer
In this paper we show how group theoretic information can be used to derive a set of necessary conditions on the coefficients of L(y) for L(y) = 0 to have a liouvillian solution. The method is used...
Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations (1993)
Michael F. Singer, Felix Ulmer
this paper we show that the index of a 1-reducible subgroup of the differential Galois group of an ordinary homogeneous linear differential equation L(y) = 0 yields the best possible bound for the...
[summary by Jacques-Arthur Weil]
Let k be a differential field (e.g. k = Q(x) or k = C (x)) with derivation d dx. We review the methods of differential Galois theory used for solving the equation L(y) = a n y (n)
Computing the Galois Group of a Polynomial Using Linear Differential Equations
Olivier Cormier, Campus De Beaulieu, Michael F. Singer, Felix Ulmer, ...
In this paper we show how to compute the Galois group G of a polynomial f 2 Q(x)[Y ] by factoring the associated linear differential equation Lf (Y ) = 0 (and constructions of it) of minimal order...