On $abc$ and discriminants
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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.References
- Jerzy Browkin, The $abc$-conjecture, Number theory, Trends Math., Birkhäuser, Basel, 2000, pp. 75–105. MR 1764797
- Henri Darmon and Andrew Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bull. London Math. Soc. 27 (1995), no. 6, 513–543. MR 1348707, DOI 10.1112/blms/27.6.513
- Noam D. Elkies, $ABC$ implies Mordell, Internat. Math. Res. Notices 7 (1991), 99–109. MR 1141316, DOI 10.1155/S1073792891000144
- Machiel van Frankenhuysen, A lower bound in the $abc$ conjecture, J. Number Theory 82 (2000), no. 1, 91–95. MR 1755155, DOI 10.1006/jnth.1999.2484
- Gary Cornell, Joseph H. Silverman, and Glenn Stevens (eds.), Modular forms and Fermat’s last theorem, Springer-Verlag, New York, 1997. Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995. MR 1638473, DOI 10.1007/978-1-4612-1974-3
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
- Andrew Granville and H. M. Stark, $abc$ implies no “Siegel zeros” for $L$-functions of characters with negative discriminant, Invent. Math. 139 (2000), no. 3, 509–523. MR 1738058, DOI 10.1007/s002229900036
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552, DOI 10.1007/978-3-642-58227-1
- D. W. Masser, The discriminants of special equations, Mathematical Gazette 372 (1966), 158–160.
- Karl K. Norton, Numbers with small prime factors, and the least $k$th power non-residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739
- Andrzej Schinzel, Selected topics on polynomials, University of Michigan Press, Ann Arbor, Mich., 1982. MR 649775
- Andrzej Schinzel, On reducible trinomials, Dissertationes Math. (Rozprawy Mat.) 329 (1993), 83. MR 1254093
- Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
- Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. MR 1002324, DOI 10.1007/978-3-663-14060-3
- A. L. Smirnov, Hurwitz inequalities for number fields, Algebra i Analiz 4 (1992), no. 2, 186–209 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 357–375. MR 1182400
- C. L. Stewart and R. Tijdeman, On the Oesterlé-Masser conjecture, Monatsh. Math. 102 (1986), no. 3, 251–257. MR 863221, DOI 10.1007/BF01294603
- Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
Additional Information
- D. W. Masser
- Affiliation: Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland
- MR Author ID: 121080
- Email: masser@math.unibas.ch
- Received by editor(s): June 4, 2001
- Published electronically: April 17, 2002
- Communicated by: David E. Rohrlich
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1912990