Suppose I have a polynomial $f$ in $p,t,$ say $f=1 \pm p^{a_1}t^{b_1} \pm \cdots \pm p^{a_s}t^{b_s}$ with $a_i,b_i \in \mathbb{N}\cup0$ and $b_i$ is non-zero. Let $X:=${$ \ (1-p^it^j) | i,j \in \mathbb{N}\cup 0 , j \neq 0 $.} Does there exist an algorithm to determine if there exist $x_1,\ldots, x_t,y_1, \ldots, y_v \in X$ such that $f\cdot x_1 \cdot \ldots \cdot x_t=y_1 \cdot \ldots \cdot y_v$? Essentially, I want to express $f$ as a product of $p$-local shifted Riemann zeta functions and reciprocals of them.

As an example of what I mean, if $f=1+pt$ then $f\cdot(1-pt)=1-p^2t^2$ and thus $f=\frac{1-p^2t^2}{1-pt}.$