Sparsity of curves and additive and multiplicative expansion of rational maps over finite fields
Volume 188 / 2019
Acta Arithmetica 188 (2019), 401-411
MSC: Primary 11D79, 11G20, 12D10, 30C15.
DOI: 10.4064/aa180307-20-8
Published online: 12 April 2019
Abstract
For a prime $p$ and a polynomial $F(X,Y)$ over the finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions of $$ F(x,y)=0, \quad x\in\mathcal{A}, \, y\in \mathcal{B}, $$ where $\mathcal A$ and $\mathcal B$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated by compositions of several rational functions are not contained in any short intervals or small subgroups.
