Perron transforms and Hironaka's game
Introduction
The algorithm of Perron appears as an important tool in various proofs of local uniformization for valuations centered on algebraic varieties. For instance, in [9], Zariski applies this algorithm in the proof of the Local Uniformization Theorem for places of algebraic function fields over base fields of characteristic 0. Then, in [10], he uses this theorem to prove resolution of singularities for algebraic surfaces (in characteristic 0). The local uniformization problem over base fields of any characteristic is still open. In [7], Knaf and Kuhlmann use a similar algorithm in the proof that Abhyankar places admit local uniformization in any characteristic. Also, in [2], Cutkosky and Mourtada use a version of Perron transforms in the proof that reduction of the multiplicity of a characteristic hypersurface singularity along a valuation is possible if there is a finite linear projection which is defectless.
The Hironaka's game was proposed by Hironaka in [5] and [6]. This game encodes the combinatorial part of the resolution a given singularity. Different winning strategies for this game allow different resolutions for that singularity. The existence of a winning strategy for Hironaka's game was first proved by Spivakovsky in [8]. An alternative solution was presented in [11]. In [4], Hauser presents a detailed relation between Hironaka's game and its applications on resolution of singularities.
The main goal of this paper is to explicitly relate Perron transforms and Hironaka's game. Our main result (Theorem 5.2), which is given in terms of matrices with non-negative integer entries, implies the existence of a winning strategy for the Hironaka's game and also the existence of Perron transforms with some required properties.
This paper is divided as follows. In Section 2, we present and prove our main theorem. In Section 3, we present and prove Lemma 4.2 of [7] (Theorem 3.3 below). Knaf and Kuhlmann use this result as an important step to prove that every Abhyankar valuation admits local uniformization. A proof of Theorem 3.3 can be found in [3], but we show here that it follows easily from Theorem 2.1. In Section 4, we present and prove Lemma 4.1 of [2] (Theorem 4.1 below). In [2], the authors refer to a proof of it in [1]. That proof is based on the original algorithm of Zariski to prove local uniformization. Again, we show that Theorem 4.1 follows from Theorem 5.2. In section 5, we present the Hironaka's game (also known as Hironaka's polyhedra game), and deduce from Theorem 2.1 that it admits a wining strategy (Theorem 5.2).
Section snippets
Main theorem
Let J be a non-empty subset of and . We define the matrix by Notice that and, as a linear map,
The main result of this paper is the following: Theorem 2.1 Let . Then there exist and finite sequences and such that, for , and componentwise, where
Kuhlmann and Knaf's Perron transform
Let Γ be a finitely generated ordered abelian group and a basis of Γ (i.e., ) formed by positive elements. Such basis exists because every ordered abelian group is free; see [3]. Definition 3.1 A simple Perron transform on is a new basis of Γ, obtained in the following way: let and such that for all . Then
Cutkosky and Mourtada's Perron transform
Let be a polynomial ring over a field k. Let ν be a valuation in with center , that is, and for all . Suppose that is a rational basis of , where is the value group of ν. Let be such that where , and for every . In [2], the authors call the inclusion map a Perron transform of type (6).
Theorem 4.1 Let and Lemma 4.1 of [2]
Hironaka's game
Let V be a finite number of points in with positive convex hull . Consider two “players”, and , competing in the following game (known as Hironaka's polihedra game): player chooses a subset J of and, afterwards, player chooses and element . After this “round”, the set V is replaced by the set obtained as follows: for each element the corresponding element will be We define then
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During the realization of this project the second author was supported by a grant from Fundação de Amparo à Pesquisa do Estado de São Paulo (process number 2017/17835-9).