The ABC theorem for higher-dimensional function fields

Hsia, Liang Chung; Wang, Julie Tzu Yueh

doi:10.1090/s0002-9947-03-03363-4

Cited by 12 publications

(6 citation statements)

References 21 publications

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“…, f n ) the local hyper-Wronskian at ν. By Proposition 2.1 in [11], for two local parameters t, u, we have a formula…”

Section: Gcd-estimates In Arbitrary Characteristicmentioning

confidence: 99%

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Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields

Corvaja¹,

Zannier²

2013

J. Eur. Math. Soc.

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In our previous work [4] we proved a bound for gcd(u − 1, v − 1), for S-units u, v of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in [3]. As pointed out by Silverman [15], the exact analogue does not work for function fields in positive characteristic. In the present work, we investigate possible extensions in that direction; it turns out that under suitable assumptions some of the results still hold. For instance we prove Theorems 2 and 3 below, from which we deduce in particular a new proof of Weil's bound for the number of rational points on a curve over finite fields (see §4). When the genus of the curve is large compared to the characteristic, we can even go beyond it. What seems a new feature is the analogy with the characteristic zero case, which admitted applications to apparently distant problems.

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“…, f n ) the local hyper-Wronskian at ν. By Proposition 2.1 in [11], for two local parameters t, u, we have a formula…”

Section: Gcd-estimates In Arbitrary Characteristicmentioning

confidence: 99%

“…Now the proof will proceed by taking suitable hyper-Wronskians. We follow the treatment of Garcia-Voloch [8] and Hsia-Wang [11]. For a separating element t ∈ L, we define as in §1 of [8] a sequence of differential operators D n,t = D n , for n = 0, 1, 2, .…”

Section: Gcd-estimates In Arbitrary Characteristicmentioning

confidence: 99%

Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields

Corvaja¹,

Zannier²

2013

J. Eur. Math. Soc.

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“…The Stothers-Mason theorem has been generalized in many different directions, for instance, to sums in one-dimensional function fields by Mason [28], by Voloch [43] and by Brownawell and Masser [2], to sums of pairwise relatively prime polynomials of several variables by Shapiro and Sparer [32], to sums in higher-dimensional function fields by Hsia and Wang [19], and to quantum deformations of polynomials by Vaserstein [41]. Motivated by the analogy between Diophantine approximation and Nevanlinna theory [42], the abc theorem has also been proven for complex entire functions by Van Frankenhuysen [39,40], for p-adic entire functions by Hu and Yang [20], and for non-Archimedean entire functions of several variables by Cherry and Toropu [5].…”

Section: Difference Analogue Of the Stothers-mason Theoremmentioning

confidence: 99%

A Stothers–Mason theorem with a difference radical

Ishizaki

Korhonen

et al. 2020

Math. Z.

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Differential calculus is not a unique way to observe polynomial equations such as $$a+b=c$$ a + b = c . We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Results, for example, on the non-existence of polynomial solutions to difference Fermat and difference Super-Fermat functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider these functional equations with non-polynomial entire solutions. Equations with polynomial or non-polynomial solutions are observed to see the sharpness of results obtained.

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“…The work of Mason and Silverman has been extended in various directions. Hsia and Wang [6] looked at the equation…”

Section: Introductionmentioning

confidence: 99%