Generalized Hensel's lemma

Let (K, v) be a complete, rank-1 valued field with valuation ring R-v and r esidue 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)(Sigma(i)a(i)x(i)) = min(i){v(a(i))}. The classical Hensel's lemma asserts that if polynomials F(x), G(0)(x), H-0(x) in R-v[x] are such that (i) v(x)(F(x) - G(0)(x)H-0(x) ) > 0, (ii) the leading coefficient of G(0)(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of v(x) sa tisfying v(x)(A(x)G(0)(x)+ B(x)H-0(x) - 1) > 0, then there exist G(x), H(x) in K[x] such that; (a) F(x)= G(x)H(x), (b) degG(x) = degG(0)(x), (c) v(x)( G(x) - G(0)(x)) > 0, v(x)(H(x) - H-0(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 w a transcendental extension of k(v).