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On $abc$ and discriminants


Author: D. W. Masser
Journal: Proc. Amer. Math. Soc. 130 (2002), 3141-3150
MSC (2000): Primary 11D61, 11P99, 11S99
DOI: https://doi.org/10.1090/S0002-9939-02-06589-9
Published electronically: April 17, 2002
MathSciNet review: 1912990
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Abstract: We modify the $abc$-conjecture for number fields $K$ in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent $\mu>1$ of the absolute value $D_K$ of the discriminant cannot be replaced by $\mu=1$, and even that an arbitrarily large power of $\log D_K$ must be present.


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  • [B] J. Browkin, The $abc$-conjecture, Number Theory (eds. R. P. Bambah, V. C. Dumir, R. J. Hans-Gill), Trends in Mathematics, Birkhäuser, 2000, 75-105. MR 2001f:11053
  • [DG] H. Darmon and A. Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bull. London Math. Soc. 27 (1995), 513-543. MR 96e:11042
  • [E] N. Elkies, $ABC$ implies Mordell, Int. Math. Res. Notices 7 (1991), 99-109. MR 93d:11064
  • [Fra] M. von Frankenhuysen, A lower bound in the $abc$ Conjecture, J. Number Theory 82 (2000), 91-95. MR 2001m:11109
  • [Fre] G. Frey, On ternary equations of Fermat type and relations with elliptic curves, Modular forms and Fermat's last theorem (eds. G. Cornell, J. H. Silverman, G. Stevens), Springer, 1997, 527-548. MR 99k:11004
  • [GKZ] I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, 1994. MR 95e:14045
  • [GS] A. Granville and H. M. Stark, $ABC$ implies no ``Siegel zeros'' for $L$-functions of characters with negative discriminant, Invent. Math. 139 (2000), 509-523. MR 2002b:11114
  • [L1] S. Lang, Fundamentals of diophantine geometry, Springer, 1983. MR 85j:11005
  • [L2] S. Lang, Number theory III, Encyclopaedia of Mathematical Sciences, Vol. 60, Springer, 1991. MR 93a:11048
  • [M] D. W. Masser, The discriminants of special equations, Mathematical Gazette 372 (1966), 158-160.
  • [N] K. K. Norton, Numbers with small prime factors, and the least $k$th power non-residue, Mem. Amer. Math. Soc. 106 (1971). MR 44:3948
  • [Sc1] A. Schinzel, Selected topics on polynomials, University of Michigan, 1982. MR 84k:12010
  • [Sc2] A. Schinzel, On reducible trinomials, Dissertationes Math. CCCXXIX (1993). MR 95d:11146
  • [Se1] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Pub. Math. I.H.E.S. 54 (1981), 323-401. MR 83k:12011
  • [Se2] J.-P. Serre, Lectures on the Mordell-Weil Theorem, Aspects of Math. E15, Vieweg, 1990. MR 90e:11086
  • [Sm] A. L. Smirnov, Hurwitz inequalities for number fields, St. Petersburg Math. J. 4 (1993), 357-375. MR 93h:11065
  • [ST] C. L. Stewart and R. Tijdeman, On the Oesterlé-Masser conjecture, Monatshefte Math. 102 (1986), 251-257. MR 87k:11077
  • [V] P. Vojta, Diophantine approximations and value-distribution theory, Lecture Notes 1239, Springer, 1987. MR 91k:11049

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Additional Information

D. W. Masser
Affiliation: Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland
Email: masser@math.unibas.ch

DOI: https://doi.org/10.1090/S0002-9939-02-06589-9
Received by editor(s): June 4, 2001
Published electronically: April 17, 2002
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society

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