Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ is absolutely irreducible, and so the Lang-Weil estimate implies $$\#V(\mathbb{F}_p) = p^{\dim V} (1+O(p^{-1/2})).$$ In particular, there are $\mathbb{F}_p$-rational points on $V$ if $p$ is sufficiently large.
I want to find points with small coordinates. A more precise formulation: How small can $1\leq B(p)\leq p$ be such that there exists a rational point $(a_1,\ldots, a_n)\in V(\mathbb{F}_q)$ with all $1\leq a_i \leq B(p)$?
I am aware that in the special case of hyper-ellicptic curves it relates to the least quadratic nonresidue, to short exponential sums, and to the Burgess bound; this question asks what happens in the general case (for me it suffices to consider singular complete intersections with singular locus of co-dimension 2, but I think it shouldn't be easier than the general case).
