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

TL;DR

This paper proves that algebraic equations with certain bounded-term conditions have solutions with bounded complexity, extending classical conjectures and applying to toric varieties, integral points, and non-standard arithmetic.

Contribution

It introduces new methods to establish bounds on solutions of algebraic equations with bounded terms, generalizing previous results and connecting to toric geometry and integral points.

Findings

Bounded number of terms for solutions of algebraic equations with bounded x-terms.

Applications to integral points on toric varieties and finite covers.

A Bertini-type irreducibility theorem over algebraic multiplicative cosets.

Abstract

An old conjecture of Erd\H{o}s and R\'enyi, 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 $\mathbb{G}_{\mathrm{m}}^l$. 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 algebraic and integral-closedness statement inside the ring of non-standard polynomials.