Results 1  10
of
19
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 ..."
Abstract

Cited by 23 (7 self)
 Add to MetaCart
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.
Differential equations for algebraic functions
 ISSAC’07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
, 2007
"... Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose deg ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
(Show Context)
Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series. 1.
Computing monodromy groups defined by plane algebraic curves
 In: Proceedings of the 2007 International Workshop on Symbolicnumeric Computation. ACM, NewYork
, 2007
"... We present a symbolicnumeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
We present a symbolicnumeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution is threefold: first of all, we show how to use a minimum spanning tree to minimize the length of paths; then, we propose a strategy that gives a good compromise between the number of steps and the truncation orders of Puiseux expansions, obtaining for the first time a complexity result about the number of steps; finally, we present an efficient numericalmodular algorithm to compute Puiseux expansions above critical points, which is a non trivial task.
Analytical Solution of Linear Ordinary Differential Equations by Differential Transfer Matrix Method
 Electronic Journal of Differential Equations
"... Abstract. We report a new analytical method for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduc ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We report a new analytical method for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the nthorder differential equation to a system of n linear differential equations with unity order. The solution is then found by integration and taking the matrix exponential of a kernel matrix. We prove the validity of method by direct substitution of the solution in the original differential equation. We discuss the general properties of differential transfer matrices and present several analytical examples, showing the applicability of the method. We shall show that the AbelLiouvilleOstogradski theorem can be easily recovered through this approach. 1.
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 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
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.
Algebraic Gfunctions associated to matrices over a groupring, preprint
"... Abstract. Given a square matrix with elements in the groupring of a group, one can consider the sequence formed by the trace (in the sense of the groupring) of its powers. We prove that the corresponding generating series is an algebraic Gfunction (in the sense of Siegel) when the group is free o ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. Given a square matrix with elements in the groupring of a group, one can consider the sequence formed by the trace (in the sense of the groupring) of its powers. We prove that the corresponding generating series is an algebraic Gfunction (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic number, and their Green function is algebraic. Our proof uses the notion of rational and algebraic power series in noncommuting variables and is an easy application of a theorem of Haiman. Haiman’s theorem uses results of linguistics regarding regular and contextfree language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a Gfunction. We ask whether the latter holds for general hyperbolic groups. Contents
Algebraic General Solutions of Algebraic Ordinary Differential Equations
"... In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For an autonomous first order ODE, we give an optimized bound for the degree of its algebraic general solutions and a polynomialtime algorithm to compute an algebraic general solu ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For an autonomous first order ODE, we give an optimized bound for the degree of its algebraic general solutions and a polynomialtime algorithm to compute an algebraic general solution if it exists. 1.