I'm looking into certain type of exponential sums, which are summed over a linear subspace, and I couldn't find a good reference for that.

The (simplified) setting is the following. Let $p$ be a prime, and consider a homogeneous polynomial $f:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p$. I'm interested in sums of the form: $$ \sum_{x\in \mathcal{S}}\omega^{f(x)} $$ where $\omega=e^{2\pi i/p}$, and $\mathcal{S}\subseteq \mathbb{Z}_p^n$ is defined to be the set of solutions to the linear system of equations $\langle y_i , x \rangle$ for linearly independent $y_1,...,y_k\in \mathbb{Z}_p^n$ (assume $k<n$).

Any reference related to that would be useful (even possibly easier cases, e.g. $p=2, d=2$, would be very helpful).

Thanks!