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On fewnomials, integral points, and a toric version of Bertini's theorem

Authors: Clemens Fuchs, Vincenzo Mantova and Umberto Zannier
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 11C08; Secondary 12E05, 12Y05, 14G05, 14J99, 11U10
Published electronically: March 1, 2017
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Abstract: An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $ g(x)\in \mathbb{C}[x]$ when its square $ g(x)^2$ has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open.

In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations $ f(x,g(x))=0$, where $ f(x,y)$ is monic of arbitrary degree in $ y$ and has boundedly many terms in $ x$: we prove that the number of terms of such a $ g(x)$ is necessarily bounded. This includes the previous results as extremely special cases.

We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus $ \mathbb{G}_{\rm m}^l$. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of $ \mathbb{G}_{\rm m}^l$, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials.

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Additional Information

Clemens Fuchs
Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstrasse 34/I, A-5020 Salzburg, Austria

Vincenzo Mantova
Affiliation: School of Science and Technology, Mathematics Division, University of Camerino, Via Madonna delle Carceri 9, IT-62032 Camerino, Italy
Address at time of publication: School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom

Umberto Zannier
Affiliation: Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, IT-56126 Pisa, Italy

Received by editor(s): December 2, 2014
Received by editor(s) in revised form: October 10, 2016, and January 10, 2017
Published electronically: March 1, 2017
Additional Notes: The first author was supported by FWF (Austrian Science Fund) grant No. P24574.
The second author was supported by the Italian FIRB 2010 RBFR10V792 “New advances in the Model Theory of exponentiation.”
The authors were also supported by the ERC-AdG 267273 “Diophantine Problems.”
Article copyright: © Copyright 2017 American Mathematical Society