Generalized Hensel's lemma

Abstract

Let (K, v) be a complete, rank-1 valued field with valuation ring R_v, and residue field k_v. Let v^x be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by v^x(∑,a₁x³=min_i{v(a_i)}inline1 The classical Hensel's lemma asserts that if polynomials F(x), G₀(x), H₀(x) in R_v[x] are such that (i) v⁰(F(x) – G₀(x)H0(x)) > 0, (ii) the leading coefficient of G₀(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of v^x satisfying v_x(A(x)G₀(x) + B(x)H₀(x) – 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)H(x), (b) deg G(x) = deg G₀(x), (c) v^x(G(x)–G₀(x)) > 0, v^x(H(x) – H₀(x)) > 0. In this paper, our goal is to prove an analogous result when v^x is replaced by any prolongation w of v to K(x), with the residue field of wa transcendental extension of k_v.