@TerryTao Thanks for the reference. I was thinking more in the regime of constant $d$ and $q$, but $m$ slightly growing with $n$. The dependency on the number of low-degree polynomials needed on $m$ seems quite bad in the paper.

@BorisBukh I meant, do we have good lower bounds on the number of common zeros of a system of low degree homogeneous polynomials over finite fields? If the degree of the polynomials is one, then the number of solutions is $q^{n-r}$, where $r$ is the rank of the system. Are there similar statements for higher degree homogeneous polynomials?