# On the construction of valuations and generating sequences on hypersurface singularities

## TL;DR

This paper develops an algorithm to describe the structure of associated graded rings of hypersurface singularities in valued fields, addressing complexities like ramification and defect, and providing finite generators in many cases.

## Contribution

It introduces a new algorithm for constructing generating sequences and analyzing associated graded rings of hypersurface singularities, especially in positive characteristic.

## Key findings

- Algorithm produces finite generators for associated graded rings in many cases.
- In rank-one valuations without defect, the algorithm yields a generating sequence in a local extension.
- The method interacts with phenomena like ramification, non-tameness, and defect, impacting local uniformization.

## Abstract

Suppose that (K, $\nu$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $\omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $\nu$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $\omega$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $\omega$ as an extension of the associated graded ring of A for the filtration defined by $\nu$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $\omega$ A[z]/(f (z)) generate it as a gr $\nu$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $\nu$) $\subset$ (L, $\omega$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $\nu$ if and only if there is no defect. In this case, gr $\omega$ A1[z]/(f (z)) is a finitely presented gr $\nu$ A1-module. This is an improved version, thanks to a referee's remarks.