Would Evaluating a polynomial at uniformly random points outputs random values? I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?

In a comment it is specified that we are working in a finite field. Here is a simple counterexample. 
Consider the field of $3$ elements, and the polynomial $p(x)=x^2$. If we evaluate at points in the field chosen at random, we get that $p(x)=0$ with probability $\frac{1}{3}$, and $p(x)=1$ with probability $\frac{2}{3}$.
