Applications of resultant of two $p$-adic power series

Abstract

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 $\phi$.