# An $abcd$ theorem over function fields and applications

Pietro Corvaja; Umberto Zannier

Bulletin de la Société Mathématique de France (2011)

- Volume: 139, Issue: 4, page 437-454
- ISSN: 0037-9484

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topCorvaja, Pietro, and Zannier, Umberto. "An $abcd$ theorem over function fields and applications." Bulletin de la Société Mathématique de France 139.4 (2011): 437-454. <http://eudml.org/doc/272636>.

@article{Corvaja2011,

abstract = {We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves, provided $a\ge 10^4$ and $c\ge 2$; this provides special cases of a known conjecture of Bogomolov.},

author = {Corvaja, Pietro, Zannier, Umberto},

journal = {Bulletin de la Société Mathématique de France},

keywords = {abc conjecture; function fields; curves on algebraic surfaces; S-units},

language = {eng},

number = {4},

pages = {437-454},

publisher = {Société mathématique de France},

title = {An $abcd$ theorem over function fields and applications},

url = {http://eudml.org/doc/272636},

volume = {139},

year = {2011},

}

TY - JOUR

AU - Corvaja, Pietro

AU - Zannier, Umberto

TI - An $abcd$ theorem over function fields and applications

JO - Bulletin de la Société Mathématique de France

PY - 2011

PB - Société mathématique de France

VL - 139

IS - 4

SP - 437

EP - 454

AB - We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves, provided $a\ge 10^4$ and $c\ge 2$; this provides special cases of a known conjecture of Bogomolov.

LA - eng

KW - abc conjecture; function fields; curves on algebraic surfaces; S-units

UR - http://eudml.org/doc/272636

ER -

## References

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- [6] P. Corvaja & U. Zannier – « Some cases of Vojta’s conjecture on integral points over function fields », J. Algebraic Geom. 17 (2008), p. 295–333. Addendum in Asian Journal of Math., 14 (2010), p. 581–584. Zbl1221.11146MR2369088
- [7] O. Debarre – Higher-dimensional algebraic geometry, Universitext, Springer, 2001. Zbl0978.14001MR1841091
- [8] A. Dujella, C. Fuchs & F. Luca – « A polynomial variant of a problem of Diophantus for pure powers », Int. J. Number Theory4 (2008), p. 57–71. Zbl1218.11028MR2387916
- [9] U. Zannier – « Some remarks on the $S$-unit equation in function fields », Acta Arith.64 (1993), p. 87–98. Zbl0786.11019MR1220487

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