@unpublished {2014,
title = {On fewnomials, integral points and a toric version of Bertini{\textquoteright}s theorem},
year = {2014},
month = {2014/12},
pages = {43},
type = {preprint},
abstract = {An old conjecture of Erd{\H o}s and R{\'e}nyi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g(x){\i}n\mathbb{C}[x]$ 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 $\mathbb{G}_{\mathrm{m}}^l$. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of $\mathbb{G}_{\mathrm{m}}^l$, 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 integral-closedness statement inside the ring of non-standard polynomials. As a byproduct of the proofs, we also provide a treatment of Puiseux expansions in several variables which seems to be missing in the existing literature.},
url = {http://arxiv.org/abs/1412.4548},
author = {Fuchs, Clemens and Mantova, Vincenzo and Zannier, Umberto}
}