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

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.