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29
Short Programs for functions on Curves
 IBM THOMAS J. WATSON RESEARCH CENTER
, 1986
"... The problem of deducing a function on an algebraic curve having a given divisor is important in the field of indefinite integration. Indeed, it is the main computational step in determining whether an algebraic function posseses an indefinite integral. It has also become important recently in th ..."
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The problem of deducing a function on an algebraic curve having a given divisor is important in the field of indefinite integration. Indeed, it is the main computational step in determining whether an algebraic function posseses an indefinite integral. It has also become important recently in the study of discrete elliptic logarithms in cryptography, and in the construction of the new class of errorcorrecting codes which exceed the VarshamovGilbert bound. It can also be used to give a partial answer to a question raised by Schoof in his paper on computing the exact number of points on an elliptic curve over a finite field. Heretofore,
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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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
Undecidability and incompleteness in classical mechanics
 Internat. J. Theoret. Physics
, 1991
"... We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary realvalued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamilt ..."
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Cited by 16 (4 self)
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We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary realvalued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of G6del's incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained. 1.
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...
The Calculation of Radical Ideals in Positive Characteristic
 Journal of Symbolic Computation
, 2002
"... We propose an algorithm for computing the radical of a polynomial ideal in positive characteristic. The algorithm does not involve polynomial factorization. ..."
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We propose an algorithm for computing the radical of a polynomial ideal in positive characteristic. The algorithm does not involve polynomial factorization.
The Inverse Problem in Differential Galois Theory
, 1996
"... this paper will be assumed to be of characteristic zero. the subscript k and refer to C). A linear differential equation is an equation ..."
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Cited by 9 (1 self)
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this paper will be assumed to be of characteristic zero. the subscript k and refer to C). A linear differential equation is an equation
Scratchpad's View of Algebra I: Basic Commutative Algebra
, 1990
"... . While computer algebra systems have dealt with polynomials and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent problem. In this paper we explain how one system solves this problem, what types and operators ..."
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Cited by 4 (0 self)
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. While computer algebra systems have dealt with polynomials and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent problem. In this paper we explain how one system solves this problem, what types and operators it is necessary to introduce and, in short, how one can construct a computational theory of commutative algebra. Of necessity, such a theory is rather different from the conventional, nonconstructive, theory. It is also somewhat different from the theories of Seidenberg [1974] and his school, who are not particularly concerned with practical questions of efficiency. Introduction This paper describes the constructive theory of commutative algebra which underlies that part of Scratchpad which deals with commutative algebra. We begin by explaining the background that led the Scratchpad group to construct such a general theory. We contrast the general theory in Scratchpad with Reduce3's the...
Computing the Abel map
, 2007
"... A Riemann surface is associated with its Jacobian through the Abel map. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of the Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singular ..."
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A Riemann surface is associated with its Jacobian through the Abel map. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of the Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singularities, the Abel map of any connected compact Riemann surface may be obtained in this way. 1
The AXIOM System
"... . AXIOM* is a computer algebra system superficially like many others, but fundamentally different in its internal construction, and therefore in the possibilities it offers to its users. In these lecture notes, we will ffl outline the highlevel design of the AXIOM kernel and the AXIOM type system, ..."
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. AXIOM* is a computer algebra system superficially like many others, but fundamentally different in its internal construction, and therefore in the possibilities it offers to its users. In these lecture notes, we will ffl outline the highlevel design of the AXIOM kernel and the AXIOM type system, ffl explain some of the algebraic facilities implemented in AXIOM, which may be more general than the reader is used to, ffl show how the type system and the information system interact, ffl give some references to the literature on particular aspects of AXIOM, and ffl suggest the way forward. A little history In 1978 the present author spent two months at IBM Yorktown Heights, as part of the Computer Algebra Group, which had developed the Scratchpad1 computer algebra system. Though this system never saw the light of day outside IBM, it was at the time a competitor for Macsyma and Reduce. All systems had struggled with the problems of writing ever more complicated algebraic algorithm...
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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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.