Th\'eor\`eme de Kaplansky effectif pour des valuations de rang 1 centr\'ees sur des anneaux locaux r\'eguliers et complets

TL;DR

This paper demonstrates that complete regular local rings with rank 1 valuations can be embedded into rings of generalized Puiseux expansions, providing a new structural insight into such rings.

Contribution

It establishes a Kaplansky-type theorem for valuations of rank 1 on complete regular local rings, extending the understanding of their embeddings.

Findings

Complete regular local rings with rank 1 valuations can be embedded into generalized Puiseux expansion rings.

The result generalizes classical valuation embedding theorems to a broader class of rings.

Provides a new tool for analyzing the structure of valuations on regular local rings.

Abstract

We prove that any complete regular local ring with a valuation of rank 1 can be embedded, as a valued ring, in a ring of generalized Puiseux expansions.