Factorization of polynomials over valued fields based on graded polynomials

In this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal factorization.

Newtonianity of Directed Unions

This chapter considers the newtonianity of directed unions and proves an analogue of Hensel's Lemma for ω‎-free differential-valued fields of H-type: Theorem 15.0.1. Here K is an H-asymptotic field with asymptotic couple (Γ‎, ψ‎), and γ‎ ranges over Γ‎. The chapter first describes finitely many exceptional values, integration and the extension K(x), and approximating zeros of differential polynomials before proving Theorem 15.0.1, which states: If K is d-valued with ∂K = K, and K is a directed union of spherically complete grounded d-valued subfields, then K is newtonian. In concrete cases the hypothesis K = ∂K in the theorem can often be verified by means of Corollary 15.2.4.

THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.