If $P_1$, $P_2$, ..., $P_m$ are $n$-variate homogeneous polynomials of degree d over a finite field $F_q$, where $q$ is much larger than $d$, but much smaller than $n$, then do we know good lower bounds on the number of common zeros of $P_1$, $P_2$, ..., $P_m$?

If $m$ is equal to $1$, then we would get some non-trivial lower bounds from the Weil bounds. But what happens for a general $m < n$? Are there any accessible references?