  In this paper, we study the hardness of solving graph-structured linear
systems with coefficients over a finite field $\mathbb{Z}_p$ and over a
polynomial ring $\mathbb{F}[x_1,\ldots,x_t]$.
  We reduce solving general linear systems in $\mathbb{Z}_p$ to solving
unit-weight low-degree graph Laplacians over $\mathbb{Z}_p$ with a
polylogarithmic overhead on the number of non-zeros. Given the hardness of
solving general linear systems in $\mathbb{Z}_p$ [Casacuberta-Kyng 2022], this
result shows that it is unlikely that we can generalize Laplacian solvers over
$\mathbb{R}$, or finite-element based methods over $\mathbb{R}$ in general, to
a finite-field setting. We also reduce solving general linear systems over
$\mathbb{Z}_p$ to solving linear systems whose coefficient matrices are walk
matrices (matrices with all ones on the diagonal) and normalized Laplacians
(Laplacians that are also walk matrices) over $\mathbb{Z}_p$.
  We often need to apply linear system solvers to random linear systems, in
which case the worst case analysis above might be less relevant. For example,
we often need to substitute variables in a symbolic matrix with random values.
Here, a symbolic matrix is simply a matrix whose entries are in a polynomial
ring $\mathbb{F}[x_1, \ldots, x_t]$. We formally define the reducibility
between symbolic matrix classes, which are classified in terms of the degrees
of the entries and the number of occurrences of the variables. We show that the
determinant identity testing problem for symbolic matrices with polynomial
degree $1$ and variable multiplicity at most $3$ is at least as hard as the
same problem for general matrices over $\mathbb{R}$.
