The distribution of rational points and polynomial maps on an affine variety over a finite field on average

الأصل · EN

Let p be a prime, let V/Fₚ be an absolutely irreducible affine variety inside the affine r-space. In this paper, we consider the problem of how often a box B will contain the expected number of points. In particular, we give a lower bound on the volume of B that guarantees almost all translations of B in the r-space contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense.