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

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 $\mathbb {G}_\textrm {m}^l$. 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.