Geodesic
On fewnomials, integral points, and a toric version of Bertini’s theorem
Fuchs, Clemens1 ; Mantova, Vincenzo2, 3 ; Zannier, Umberto4
1 Department of Mathematics, University of Salzburg, Hellbrunnerstrasse 34/I, A-5020 Salzburg, Austria
2 School of Science and Technology, Mathematics Division, University of Camerino, Via Madonna delle Carceri 9, IT-62032 Camerino, Italy
3 School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom
4 Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, IT-56126 Pisa, Italy
Journal of the American Mathematical Society, Tome 31 (2018) no. 1, pp. 107-134

Voir la notice de l'article provenant de la source American Mathematical Society

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 $\mathbb {G}_\textrm {m}^l$. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of $\mathbb {G}_\textrm {m}^l$, 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.
DOI : 10.1090/jams/878

Fuchs, Clemens 1 ; Mantova, Vincenzo 2, 3 ; Zannier, Umberto 4

1 Department of Mathematics, University of Salzburg, Hellbrunnerstrasse 34/I, A-5020 Salzburg, Austria
2 School of Science and Technology, Mathematics Division, University of Camerino, Via Madonna delle Carceri 9, IT-62032 Camerino, Italy
3 School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom
4 Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, IT-56126 Pisa, Italy 
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Fuchs, Clemens; Mantova, Vincenzo; Zannier, Umberto. On fewnomials, integral points, and a toric version of Bertini’s theorem. Journal of the American Mathematical Society, Tome 31 (2018) no. 1, pp. 107-134. doi: 10.1090/jams/878

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