Let $n\in\mathbb{N}$ and let $q\in[n,2n]$ be a prime number.
In addition, let s,s':\mathbb{F}_q\to\mathbb{F}_q be polynomials of degree $\sqrt{n}$ such that s\neq s'.
From the Schwartz–Zippel lemma (or simply because s-s' is also a polynomials of degree $\sqrt{n}$, thus is has at most $\sqrt{n}$ roots) we know that \Pr_{r\in \mathbb{F}_q}[s(r)=s'(r)]\leq\frac{\sqrt{n}}{q}\leq\frac{1}{\sqrt{n}}
What is the best upper bound I can give for the probability over $r$ which is uniformly picked out of $\{1,\ldots,\sqrt{n}\}$ ? That is \Pr_{r\in \{1,\ldots,\sqrt{n}\}}[s(r)=s'(r)]\leq\quad?