Desingularization of function fields

TL;DR

This paper presents an algebraic method for desingularizing fields of fractions of certain domains, explicitly describing valuations with local parameters and units, using a rooted tree structure based on valuations and monomial orderings.

Contribution

It introduces a novel algebraic framework for desingularization of function fields, emphasizing explicit valuation descriptions and a tree-based structure for arbitrary dimensions and characteristics.

Findings

Provides a rooted tree model for desingularization

Explicitly describes valuations using local parameters

Applicable to arbitrary dimension and characteristic

Abstract

This is a self-contained purely algebraic treatment of desingularization of fields of fractions $\mathbf{L}:=Q(\mathbf{A})$ of $d$-dimensional domains of the form \[\mathbf{A}:=\bar{\mathbf{F}}[\underline{x}]/\langle b(\underline{x})\rangle\] with a purely algebraic objective of uniquely describing $d$-dimensional valuations in terms of $d$ explicit (independent) local parameters and $1$ (dependent) local unit, for arbitrary dimension $d$ and arbitrary characteristic $p$. The desingularization will be given as a rooted tree with nodes labelled by domains $\mathbf{A}_k$ (all with field of fractions $Q(\mathbf{A}_k)=\mathbf{L}$), sets $EQ_k$ and $INEQ_k$ of equality constraints and inequality constraints, and birational change-of-variables maps on $\mathbf{L}$. The approach is based on d-dimensional discrete valuations and local monomial orderings to emphasize formal Laurent series expansions in $d$ independent variables. It is non-standard in its notation and perspective.