Points on curves in small boxes and applications (Q464661)

Let \(\mathbb{F}_p\) denote the finite field of \(p\) elements and let \(1\leq M < p\). The authors obtain upper bounds on the number of solutions of the congruences NEWLINE\[NEWLINE f(x) \equiv y \pmod p \quad \text{ and } \quad f(x) \equiv y^2 \pmod p,NEWLINE\]NEWLINE for a polynomial \(f \in \mathbb{F}_p[X]\), where \((x,y) \in [R+1, R+M]\times [S+1,S+M]\). Denoting the number of solutions to the first congruence as \(J_f(M;R,S)\), it is shown (Theorem 5) that NEWLINE\[NEWLINE J_f(M;R,S) \ll \frac{M^2}{p} + M^{1-1/2^{m-1}} p^{o(1)},NEWLINE\]NEWLINE for an arbitrary polynomial of degree \(m\geq 2\) as \(p \rightarrow \infty\). Furthermore, the authors prove detailed results for the number of solutions to the second congruence, denoted by \(I_f(M;R,S)\). The behavior of \(I_f(M;R,S)\) for \(M \rightarrow \infty\) and for polynomials of degree 3 is neatly summarized in Corollary 3, whereas Theorem 4 gives asymptotic bounds for \(I_f(M;R,S)\) for polynomials of degree \(m\geq 4\).NEWLINENEWLINEThe authors also investigate two applications of their results in Section 3. First, they study the distribution of isomorphism classes of hyperelliptic curves of genus \(\geq 1\) in thin families. Second, they consider the diameter of polynomial dynamical systems.