Dual Graphs and Generating Sequences of Non-divisorial Valuations on Two-dimensional Function Fields

TL;DR

This paper explores dual graphs of valuations in two-dimensional function fields, providing explicit generating sequences for non-divisorial valuations using elementary number theory methods.

Contribution

It introduces a new definition of generating sequences and explicitly formulates their values for non-divisorial valuations based on dual graph data.

Findings

Explicit formulas for generating sequences of non-divisorial valuations

Elementary proofs using continued fractions and Frobenius problem

Clarification of the relationship between dual graphs and valuation sequences

Abstract

An exposition on Spivakovsky's dual graphs of valuations on function fields of dimension two is first given, leading to a proof of minimal generating sequences for the non-divisorial valuations. It should be noted that the definition of generating sequence used in this paper is different from Spivakovsky's original usage. This change leads to an explicit formulation of generating sequence values for the non-divisorial cases in terms of data from their dual graphs. The proofs are elementary in the sense that only continued fractions and the linear Diophantine Frobenius problem from classical number theory are used.