Journal Article10.1017/S0305004100061235
The S-unit equation over function fields
Joseph H. Silverman
- 01 Jan 1984
- Vol. 95, Iss: 1, pp 3-4
TL;DR: In this paper, the Riemann-Hurwitz formula has been used to give an independent proof of Mason's bound, relying only on elementary algebraic geometry and not on the Baker's deep results on linear forms in logarithms.
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Abstract: In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equationin S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.
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Citations
•Book
The Arithmetic of Elliptic Curves
Joseph H. Silverman
- 01 Jan 1986
TL;DR: It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
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•Book
Unit Equations in Diophantine Number Theory
Jan-Hendrik Evertse,Kálmán Győry +1 more
- 30 Dec 2015
TL;DR: The first volume devoted to unit equations in Diophantine number theory is as discussed by the authors, where the authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equation over finitely generated domains.
134
Uniform bounds for the number of solutions to Yn = f(X)
Jan-Hendrik Evertse,Joseph H. Silverman +1 more
- 01 Jan 1986
TL;DR: In this article, it was shown that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K, unless (n, nk) is a permutation of one of the n-tuples.
65
ABC implies primitive prime divisors in arithmetic dynamics
TL;DR: In this paper, it was shown that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that vp(ϕ n (α)) > 0a ndvp( ϕ m (α)f 0 for all positive integers m 0 with the stronger condition vp n (β) = 1.
65
Some remarks on the S-unit equation in function fields
TL;DR: In this paper, it was shown that H(u1,..., un) ≤ 4n−2(#S + 2g − 2) where S is the set of places of K where some ui is not a unit.
References
The hyperelliptic equation over function fields
R. C. Mason
- 01 Mar 1983
TL;DR: Baker as discussed by the authors showed that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients.
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