Let $J_{s,k}(N)$ denote the number of integer solutions to the system of equations $ x_1^j+\cdots x_s^j = x_{s+1}^j+\cdots+x_{2s}^j, \quad j=1,\dots,k$

with $1\le x_i\le N$ for $i=1,\dots,2s$.

We will discuss one -of the many- applications of Vinogradov's mean value theorem that makes use of estimates for the quantity $J_{s,k}(N)$ to bound the number of solutions to certain polynomial congruences in short intervals.

More concretely, we will focus on equations of the form

\[ y^2 \equiv f(x) \pmod p, \quad (x,y)\in [R+1,R+N]\times[S+1,S+N],\]

where $f\in \mathbb F_p[x]$ is a polynomial of degree $k\ge 3$ and discuss the methods available to bound the number of solutions to the previous congruence for different ranges of $N$.

In particular, to obtain non-trivial estimates for small values of $N$ the best strategy seems to be to translate the congruence into a problem over the integers.

Joint work with M.C. Chang, J. Cilleruelo, M. Z. Garaev, J. Hernandez and I. E. Shparlinski.