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On some extensions of the Ailon–Rudnick theorem

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Abstract

In this paper we present some extensions of the Ailon–Rudnick theorem, which says that if \(f,g\in \mathbb {C}[T]\), then \(\gcd (f^n-1,g^m-1)\) is bounded for all \(n,m\ge \) 1. More precisely, using a uniform bound for the number of torsion points on curves and results on the intersection of curves with algebraic subgroups of codimension at least 2, we present two such extensions in the univariate case. We also give two multivariate analogues of the Ailon–Rudnick theorem based on Hilbert’s irreducibility theorem and a result of Granville and Rudnick about torsion points on hypersurfaces.

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Acknowledgments

The author is very grateful to Joseph Silverman for drawing the attention on the Ailon–Rudnick theorem and related results. The author would also like to thank Igor Shparlinski, Joseph Silverman, Thomas Tucker and Umberto Zannier for their valuable suggestions and stimulating discussions, and also for their comments on an early version of the paper. The author is also grateful to the anonymous referee for spotting an error in the previous version of Lemma 2.8, and for other comments which improved the presentation of the paper.The research of A. O. was supported by the UNSW Vice Chancellor’s Fellowship.

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  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Alina Ostafe

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  1. Alina Ostafe
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Correspondence to Alina Ostafe.

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Communicated by A. Constantin.

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Ostafe, A. On some extensions of the Ailon–Rudnick theorem. Monatsh Math 181, 451–471 (2016). https://doi.org/10.1007/s00605-016-0911-3

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