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Arithmetic On Superelliptic Curves
 Math. Comp
, 2000
"... This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y n = c(x) which has only one point at infinity. Divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique repre ..."
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Cited by 37 (4 self)
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This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y n = c(x) which has only one point at infinity. Divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.
Computing RiemannRoch spaces in algebraic function fields and related topics
, 2001
"... this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any ..."
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Cited by 21 (0 self)
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this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any "computable" constant field k of any characteristic as long as the required integral closures can be computed, and does not involve constant field extensions
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
A polynomialtime complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function
 MR 2000j:14098
"... Dedicated to Wolfgang Schmidt on the occasion of his sixtieth birthday. Abstract. In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bitcompl ..."
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Dedicated to Wolfgang Schmidt on the occasion of his sixtieth birthday. Abstract. In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bitcomplexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein’s theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function. 1.
A quantitative version of Runge’s theorem on diophantine equations
 ACTA ARITHMETICA
, 1992
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Lattice Basis Reduction in Function Fields
 In ANTS3 : Algorithmic
, 1998
"... We present an algorithm for lattice basis reduction in function fields. In contrast to integer lattices, there is a simple algorithm which provably computes a reduced basis in polynomial time. Moreover, this algorithm works only with the coefficients of the polynomials involved, so there is no polyn ..."
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Cited by 6 (2 self)
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We present an algorithm for lattice basis reduction in function fields. In contrast to integer lattices, there is a simple algorithm which provably computes a reduced basis in polynomial time. Moreover, this algorithm works only with the coefficients of the polynomials involved, so there is no polynomial arithmetic needed. This algorithm can be generically extended to compute a reduced lattice basis starting from a generating system. Moreover, it can be applied to lattices of integral determinant over the field of puiseux expansions of a function field. In that case, this algorithm can be used for computing in Jacobians of curves. 1 Previous work In [4], A. Lenstra published a work on factoring multivariate polynomials over finite fields. Part of the problem was solved by computing a smallest vector of a lattice in a polynomial ring. To solve this problem, he formulated an algorithm which works "only" with coefficients of the finite field. The "only" means that except addition and subs...
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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Cited by 5 (0 self)
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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.
Computing All Integer Solutions of a General Elliptic Equation
, 2000
"... The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 dening elliptic curves. We explore the possibility of extending this method to include any equation f(u; v) = 0, where f 2 Z[u;v] denes an irreducible cu ..."
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Cited by 3 (3 self)
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The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 dening elliptic curves. We explore the possibility of extending this method to include any equation f(u; v) = 0, where f 2 Z[u;v] denes an irreducible curve of genus 1, independent of shape or degree of the polynomial f . We give a detailed description of the general features of our approach, putting forward along the way some claims (one of which conjectural) that are supported by the explicit examples added at the end. 1
Finding a Basis of a Linear System with Pairwise Distinct Discrete Valuations on an Algebraic Curve
 J. SYMBOLIC COMP
, 2000
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Computing All Integer Solutions of a Genus 1 Equation
"... The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u, v) = 0, where f Z[u, v] is irreducible over Q, defines a curve of genus 1 ..."
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Cited by 3 (0 self)
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The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u, v) = 0, where f Z[u, v] is irreducible over Q, defines a curve of genus 1, but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual examples corresponding to equations of degree 5 and degree 9. 1991 Mathematics subject classification: 11D41, 11G05 Key words and phrases: diophantine equation, elliptic curve, elliptic logarithm # Econometric Institute, Erasmus University, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands; email: stroeker@few.eur.nl; homepage: http://www.few.eur.nl/few/people/stroeker/ + Department of Mathematics, University of Crete, Iraklion, Greece; email: tzanakis@math.uch.gr; homepage: http://www.math.uoc.gr/tzanakis 1