Assume we have the polynomial space $R_q$ defined as $R_q = Z_q/(X^n + 1)$. We draw samples $s_i \gets R_q$ uniformly at random. Additionally, we define the error distribution $\chi$ as a discrete centred Gaussian bounded by $B$. We compute the following quantities: $h_i = s_ip_1 + e_i$ where $p_1 \in R_q$ is a fixed polynomial. We note that $s_i$ and $e_i$ are not released. The common polynomial $p_1$ is chosen randomly and is shared for all $h_i$.
Based on the hardness of RLWE problem we know that $h_i$ is computationally indistinguishable from a random sample of $R_q$. I was wondering what guarantees we have on the joint distribution $(h_0, h_1, ...)$