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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...
Diophantine equations and Bernoulli polynomials
"... Given m;n 2, we prove that, for sufficiently large y, the sum 1 n + \Delta \Delta \Delta + y n is not a product of m consecutive integers. We also prove that for m 6= n we have 1 m + \Delta \Delta \Delta + x m 6= 1 n + \Delta \Delta \Delta + y n , provided x; y are sufficiently large. ..."
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Given m;n 2, we prove that, for sufficiently large y, the sum 1 n + \Delta \Delta \Delta + y n is not a product of m consecutive integers. We also prove that for m 6= n we have 1 m + \Delta \Delta \Delta + x m 6= 1 n + \Delta \Delta \Delta + y n , provided x; y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are "almost" indecomposable, a result of independent interest.
Algebraic Geometry Over Four Rings and the Frontier to Tractability
 CONTEMPORARY MATHEMATICS
"... We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zerodimensional part of an algeb ..."
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Cited by 6 (4 self)
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We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zerodimensional part of an algebraic set over C (c) the number of connected components of a semialgebraic set We also present some results which significantly lower the complexity of deciding the emptiness of hypersurface intersections over C and Q, given the truth of the Generalized Riemann Hypothesis. Furthermore, we state some recent progress on the decidability of the prefixes 989 and 9989, quantified over the positive integers. As an application, we conclude with a result connecting Hilbert's Tenth Problem in three variables and height bounds for integral points on algebraic curves. This paper
ELLIPTIC BINOMIAL DIOPHANTINE EQUATIONS
, 1999
"... The complete sets of solutions of the equation ( n) ( m) = are k ℓ ..."
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The complete sets of solutions of the equation ( n) ( m) = are k ℓ
Some applications of Diophantine approximation
 Number Theory for the Millenium III, A.K. Peters
, 2002
"... ..."
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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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
Uncomputably Large Integral Points On Algebraic Plane Curves?
"... We show that the decidability of an amplication of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prex 989 is generically de ..."
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We show that the decidability of an amplication of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prex 989 is generically decidable. This means that we give a precise geometric classication of those polynomials f 2Z[v; x; y] for which the question 9v2N such that 8x2N 9y2N with f(v; x; y)=0? may be undecidable, and we show that this set of polynomials is quite small in a rigourous sense. (The decidability of 989 was previously an open question.) We also show that if integral points on curves can be bounded eectively, then 9989 is generically decidable as well. We thus obtain a connection between the decidability of certain Diophantine problems, height bounds for points on curves, and the geometry of certain complex surfaces and 3folds. 1.
INTEGRAL POINTS IN TWOPARAMETER ORBITS
"... Abstract. Let K be a number field, let f: P1 − → P1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w ∈ P1(K) are not preperiodic under f. We prove that the set of (m, n) ∈ N 2 such that f ◦m (u) is Sintegral relative to f ◦n (w) is ..."
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Abstract. Let K be a number field, let f: P1 − → P1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w ∈ P1(K) are not preperiodic under f. We prove that the set of (m, n) ∈ N 2 such that f ◦m (u) is Sintegral relative to f ◦n (w) is finite and effectively computable. This may be thought of as a twoparameter analog of a result of Silverman on integral points in orbits of rational maps. This issue can be translated in terms of integral points on an open subset of P 2 1; then one can apply a modern version of the method of Runge, after increasing the number of components at infinity by iterating the rational map. Alternatively, an ineffective result comes from a wellknown theorem of Vojta. 1.