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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 14 (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.
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 ..."
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Cited by 12 (4 self)
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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.
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 ..."
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Cited by 12 (5 self)
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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.
A Reduction for Regular Differential Systems
, 2003
"... We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisationfree) iteration that transforms a system with regular finite singularities into an equivalent one with s ..."
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Cited by 2 (0 self)
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We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisationfree) iteration that transforms a system with regular finite singularities into an equivalent one with simple finite poles. We then apply our iteration to systems satisfied by bases of algebraic function fields, obtaining algorithms for computing the number of irreducible components and the genus of algebraic curves.
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 ..."
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Cited by 2 (0 self)
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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 ..."
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Cited by 2 (0 self)
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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.
DIFFERENTIAL EQUATIONS FOR ALGEBRAIC FUNCTIONS
, 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 ..."
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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.
Applicable Algebra in Engineering, Communication and Computing manuscript No. (will be inserted by the editor) Complexity Bounds for the rational NewtonPuiseux Algorithm over Finite Fields
"... Abstract We carefully study the number of arithmetic operations required to compute rational Puiseux expansions of a bivariate polynomial F over a finite field. Our approach is based on the rational NewtonPuiseux algorithm introduced by D. Duval. In particular, we prove that coefficients of F may b ..."
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Abstract We carefully study the number of arithmetic operations required to compute rational Puiseux expansions of a bivariate polynomial F over a finite field. Our approach is based on the rational NewtonPuiseux algorithm introduced by D. Duval. In particular, we prove that coefficients of F may be significantly truncated and that the complexity of parts of the computation may be bounded in terms of the output size. These preliminary results lead to a more efficient version of the algorithm with a complexity upper bound that improves previously published results. This algorithm could easily be implemented in a computer algebra system; the only asymptotically “fast ” subalgorithm required to stay within our bound is the FFTbased multiplication of univariate polynomials with coefficients in a finite field.