Steffensen’s analogue for approximating roots of p-adic polynomial equations

Abstract

Hensel’s lemma is an important result in valuation theory which gives information on finding roots of polynomials. A classical application of this result deals with the problem of finding roots of a p-adic number a in the set of p-adic numbers ℚp. Lately, there are several investigations concerning the problem of finding roots of p-adic numbers using classical root-finding methods from numerical analysis. None of these previous works, however, gave account to the problem of finding roots of a general p-adic polynomial equation in ℤp[x]. Hence, motivated by this observation, we propose an analogue of Steffensen’s method in finding roots of a general polynomial equation f (x) = 0 in ℤp[x], thereby providing a generalization of previous investigations regarding root-finding problems in the p-adic setting.