Elsevier

Journal of Algebra

Volume 563, 1 December 2020, Pages 100-110
Journal of Algebra

Perron transforms and Hironaka's game

https://doi.org/10.1016/j.jalgebra.2020.05.028Get rights and content

Abstract

In this paper we present a matricial result that generalizes Hironaka's game and Perron transforms simultaneously. We also show how one can deduce the various forms in which the algorithm of Perron appears in proofs of local uniformization from our main result.

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 p>0 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 {1,,n} and jJ. We define the n×n matrixAJ,j=(ars) byars={1 if r=s or if r=j and sJ0 otherwise . Notice that detAJ,j=1 and, as a linear map,AJ,j(α1,,αn)=(α1,,αj1,iJαi,αj+1,,αn).

The main result of this paper is the following:

Theorem 2.1

Let α,βNn. Then there exist lN and finite sequences J1,,Jl{1,,n} and j1,,jl such that, for k=1,,l,Jkis chosen in function of the set{α,β,J1,,Jk1,j1,,jk1},jkis randomly assigned inJk andAαAβorAβAα componentwise, whereA=AJl,

Kuhlmann and Knaf's Perron transform

Let Γ be a finitely generated ordered abelian group and B={γ1,,γn} a basis of Γ (i.e., Γ=γ1ZγnZ) formed by positive elements. Such basis exists because every ordered abelian group is free; see [3].

Definition 3.1

A simple Perron transform on B is a new basis B1={γ1(1),,γn(1)} of Γ, obtained in the following way: let J{1,,n} and jJ such that γjγi for all iJ. Thenγi(1)={γiγjif iJ{j}γiotherwise.

Observe that B1 is indeed a basis of Γ and is formed by positive elements, since γi>γj for all iJ{j}.

Cutkosky and Mourtada's Perron transform

Let k[x1,,xm] be a polynomial ring over a field k. Let ν be a valuation in k[x1,,xm] with center (x1,,xm), that is, ν(k×)=0 and ν(f)>0 for all f(x1,,xn). Suppose that B={ν(x1),,ν(xn)} is a rational basis of ΓνQ, where Γν is the value group of ν. Let x1,,xm be such thatxi={j=1n(xj)aijif 1inxiif n<im, where aijN, det(aij)=1 and 0<ν(xi) for every 1im. In [2], the authors call the inclusion map k[x1,,xm]k[x1,,xm] a Perron transform of type (6).

Theorem 4.1

Lemma 4.1 of [2]

Let M1=x1d1xndn and M2=x1e1x

Hironaka's game

Let V be a finite number of points in Nn with positive convex hull N:=conv(V+R0n). Consider two “players”, P1 and P2, competing in the following game (known as Hironaka's polihedra game): player P1 chooses a subset J of {1,,n} and, afterwards, player P2 chooses and element jJ. After this “round”, the set V is replaced by the set V1 obtained as follows: for each element α=(a1,,an)V the corresponding element α1=(b1,,bn)V1 will bebj:=iJai and bk:=ak if kj. We define then N1:=conv(V1+R0n)

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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).

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