Suppose I have a sequence $a_n$ given by $s_1^m+s_2^m$ modulo $p$. This takes on some values which are a subset of $\mathbb{Z}_p$. Suppose I also have a second sequence $b_n$ given by $t_1^m + t_2^m$.
I would like to construct a sequence or some nice expression that describes the intersection of the values taken on by these two sequences.
We can rewrite the series as varieties over $\mathbb{Z}_p$. Let $X = s_1^m$ then $a_n$ maybe re-written as $X+X^\alpha$ (for suitable $\alpha$) and similarly for $b_n$. The question is when is $x \in \mathbb{Z}_p$ a solution to both systems.
This seems like a basic question in algebraic geometry but I don't know the subject.