Fuchs, C, Mantova, V orcid.org/0000-0002-8454-7315 and Zannier, U (2018) On fewnomials, integral points and a toric version of Bertini's theorem. Journal of the American Mathematical Society, 31 (1). pp. 107-134. ISSN 0894-0347
Abstract
An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial g(x)∈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 Glm. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of Glm, 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 American Mathematical Society. This is an author produced version of a paper accepted for publication in Journal of the American Mathematical Society. Uploaded in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 Jan 2017 10:53 |
Last Modified: | 29 Nov 2017 14:13 |
Published Version: | https://doi.org/10.1090/jams/878 |
Status: | Published |
Publisher: | American Mathematical Society |
Identification Number: | 10.1090/jams/878 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:110384 |