Narváez Macarro, Luis2025-10-282025-10-281998Narváez Macarro, L. (1998). Division Theorem over the Dwork-Monsky-Washnitzer Completion of Polynomial Rings and Weyl Algebras.0-8247-0153-4https://hdl.handle.net/11441/178310This chapter provides proofs of Weierstrass-Hironaka division theorems for the Dwork-Monsky-Washnitzer completion of polynomial and Weyl algebras in several variables with coefficients in a complete discrete valuation ring. The case of polynomial algebras generalizes the Weierstrass division theorem and precises the original proof of noetherianity. The case of Weyl algebras is considered in order to analyze division techniques over the ring of infinite linear differential operators and over a smooth weakly formal scheme. A proof in the case of the full completion of Weyl algebras in one variable is proposed independently. The chapter also discusses rings of Dwork-Monsky-Washnitzer power series, division over polynomial rings with coefficients in a field, and division over strictly convergent or power series rings.application/pdf18 p.engAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/bookPartinfo:eu-repo/semantics/openAccess