Points on curves in small boxes en applications

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x)\equiv y(\mathrm{m}\mathrm{o}\mathrm{d}p)$ and $f(x) \equiv y^2 \pmod p,$withaprime$p$andapolynomial$f$,where$(x,y)$ belongs to an arbitrary square with side length $M$. We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curves $Y^2=X^{2g+1}+a_{2g-1}{X}^{2g-1}+\mathrm{.}\mathrm{.}\mathrm{.}+a_{1}X+a_0$ over the finite field ${\text{F}}_p$ of $p$ elements, with coefficients in a $2g$-dimensional cube $ (a_0,…, a_{2g-1})\in [R_0+1,R_0+M]\times…\times [R_{2g-1}+1,R_{2g-1}+M]$ that are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over ${\text{F}}_p$.