Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can be $\mathbb{Z}_p$ and $p$ is a large prime number.
*Question: Is $\ r_1 \cdot f_1 + r_2 \cdot f_2 \ $ distributed uniformly over any subgroup of $R^d[x]$?