Applications of resultant of two p-adic power series

2021/11/09 by Tran, Hoang Anh
#11A05 #11D88 #11T30 #13P15 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2111.04974

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 Q_p, or more generally the field C_p which is the complete field of the algebraic closure Q_p with respect to the p-adic absolute value, denoted by | ⋅ |_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 ϕ(x)=min(v_p(F(x)), v_p(G(x))) over various domains, namely open and closed unit discs of K or 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 ϕ.

Related

• The Weierstrass preparation theorem and resultants of p-adic power series
• Estimating the p-adic valuation of the resultant
• Power series and p-adic algebraic closures
• Resultants and subresultants of p-adic polynomials
• Relationships between p-unit constructions for real quadratic fields