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 ) ∈ C [ x ] g(x)\in \mathbb {C}[x] when its square g ( x ) 2 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 f(x,g(x))=0 , where f ( x , y ) f(x,y) is monic of arbitrary degree in y y and has boundedly many terms in x x : we prove that the number of terms of such a g ( x ) 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 G m l \mathbb {G}_\textrm {m}^l . Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of G m l \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.