  Given a prime $p$, and $v_p(a)$ stand for the $p$-adic valuation of the
element $a$ in a finite extension $K$ of $\mathbf{Q}_p$, or more generally the
field $\mathbf{C}_p$ which is the complete field of the algebraic closure
$\mathbf{Q}_p$ with respect to the $p$-adic absolute value, denoted by $\lvert
\cdot \rvert_p$. Let $F$ and $G$ be two ($p$-adic) power series with no common
roots. We aim to estimate the maximal value $S$ and the minimal value $s$ of
the function $\phi(x)=\min(v_p(F(x)), v_p(G(x)))$ over various domains, namely
open and closed unit discs of $K$ or $\mathbf{C}_p$. To do this, we use partial
resultants of two power series over certain domains defined by varied versions
of the Weierstrass preparation theorem. Furthermore, the resultant of power
series provides an efficient tool while studying the irreducibility and
calculating the maximal value of $\phi$.
