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On some classification results of real singularities up to the arc-analytic equivalence (Singularity theory of differential maps and its applications)

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(1)

On

some

classification results of real

singularities

up

to

the

arc‐analytic

equivalence

Jean‐Uaptiste

Campesato

March

21,

2017

Abstract

\mathrm{H}\mathrm{u}\mathrm{s}noteisan

expanded

versionofatalkgiven

during

the conference

Singularity

theory ofdifferential

maps andifs

applications

attheRIMS,

Kyoto

(December6‐9,2016).

We firststatethe definition andsomepropertiesof the

arc‐analytic equivalence

which

isan

equivalence

relation withnocontinuousmodulionNash(i.e.real

analytic

andsemi‐

algebraic)

function germs. Itisa

semiaigebraic

versionof the

blow‐analytic equivalence

of T.‐C. Kuo.

Then,wepresentaninvariantof the

arc‐analytic equivalence

whichisconstructed

following

themotivic zetafunction of Denef‐Loeser.

Finally,

we

explain

howtoderive fromitsomeclassification results for Brieskom

poly‐

nomialsandmore

generally

forsome

weighted homogeneous

polynomials.

Contents

1 The

arc‐analytic

equivalence

1

2 A motivic invariantof the

arc‐analytic equivalence

2

3 Aconvolution formula 5

4

Applications:

someclassification \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\{\mathrm{s} ó

4.1

Arc‐analytic

classification of Brieskorn

polynomials

. . . 6

4.2

Arc‐analytic

classification ofsome

weighted homogeneous polynomials

. . . 8

References 9

1

The

\mathrm{a}\mathrm{r}\mathrm{c}\leftrightarrow

analytic equivalence

H.

Whitney

[25,

Example

13.1]

noticed that thecross‐ratio isacontinuousmodulusof the

family f_{t}:(\mathbb{R}^{2},0)\rightarrow \mathrm{t}\mathbb{R},0

), t\in \mathrm{t}0,1), defined

by

f_{t}(x,y)=xy(y-x)(y-tx).

Particularly,

two

distinct function germs of this

family

are never

C^{1}

‐equivalent,

i.e. if thereexists a C^{1_{-}}

diffeomorphism

$\varphi$:(\mathbb{R}^{2},0)\rightarrow(\mathbb{R}^{2},0)

such that

f_{t^{l}}=f_{t}\circ $\varphi$

then t=t'.

T.‐C. Kuo

[15]

suggested

the

blow‐analytic equivalence

as acandidatetoobtainaclas‐

sification of real

singularities

withoutcontinuousmoduli.He

proved

that thisnotion isan

equivalence

relationonreal

analytic

function germs and thatitadmitsnocontinuousmod‐

uli for isolated

singularities.

Indeed,

a

family

of real

analytic

function germs with isolated

singularities

defines

locally finitely

many

blow‐analytic equivalence

classes.

Up

tonow,the knowninvariantsof the

blow‐analytic equivalence

arethe Fukuiinvari‐

ants

[12]

and the Koike‐Parusinskizetafunctions

[14].

Inorderto constructricherinvari‐

(2)

called the blow‐Nash

equivalence.

It isarelationonNash\star function germs withnocontin‐

uousmoduli for isolated

singularities.

Thenotionof blow‐Nash

equivalence

evolved and stabilizedtothe

following:

twoNash function germs

f,g:(\mathbb{R}^{d},0)\rightarrow(\mathbb{R},0)

areblow‐Nash

equivalent

if,

after

being composed

with Nash modifications $\dagger$

,

they

are

Nash‐equivalent

viaa

Nash‐diffeomorphism

which preserves the

multiplicities

of the

Jacobian

determinants of the modifications.

Initially

itwas

expected,

butnotknown

yet,

whether this relationis

an

equivalence

relationonNashfunction germs.

The

goal

of thissection istointroducethe

arc‐analyt

\mathrm{c}

equivalence

definedin

[6].

It is

acharacterization of the blow‐Nash

equivalence

in termsof

arc‐analytic

maps. It avoids

toinvolve Nash modifications andit is an

equivalence

relation. Moreover,A.Parusirski

andL.Păunescu

[21]

recently proved

itadmitsnocontinuousmoduli,evenfor families of

non‐isolated

singularities.

Definition1.1

([6,

Definition

7.5]).

TwoNash function germs

f,g:(\mathbb{R}^{d},0)\rightarrow \mathrm{t}\mathbb{R},0

)arearc‐

analytically equivalent

if thereexistsa

semialgebraic homeomorphism

$\varphi$:(\mathbb{R}^{d},0)\rightarrow(\mathbb{R}^{d},0)

such that

(i) g=f\mathrm{o} $\varphi$,

(ii)

$\varphi$ is

arc‐analytic,

i.e. for

$\gamma$:(\mathbb{R},0)\rightarrow(\mathbb{R}^{d},0\rangle

real

analytic,

the

composition

$\varphi$\circ $\gamma$ isalso

real

analytic,

(iii)

Thereexists c>0such that

|\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{d} $\varphi$|>c

where

\mathrm{d} $\varphi$

isdefined

$\ddagger$.

Remark1.2.

By[4, Corollary3.6],

for $\varphi$asinthe

previous

definition,theconverse

$\varphi$^{-1}

isalso

arc‐analytic

and thereexists \overline{c}>0such that

|\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{d}$\varphi$^{-1}|>\tilde{c}

where

\mathrm{d}$\varphi$^{-1}

isdefined.

Particularly,

we

get

the

following

proposition.

Proposition

1.3

([6,

Proposition

7.7]).

The

arc‐analytic equivalence

isan

equivalence

relationon

Nashfunction

germs

(\mathbb{R}^{d},0)\rightarrow \mathrm{t}\mathbb{R},0

).

The

following

proposition

statesthat the

arc‐analytic equivalence

isacharacterization

of the blow‐Nash

equivalence.

Particularly,

the blow‐Nash

equivalence

isan

equivalence

relationas

expected.

Proposition

1.4

([6,

Proposition

7.9]).

Two

Nashfunction

germsare

arc‐analytically equivalent

ifand only

if

they

areblow‐Nash

equivalent.

The

following

resultensuresthat the

arc‐analytic

equivalence

hasnocontinuousmod‐

uli,evenfor families of non‐isolated

singularities.

Itisaconsequence of

[21,

Theorem

8.5]

together

with the

proof

of

[21,

Theorem

3.3]

and formula

[21, (3.9)].

Theorem1.5

(Parusi’ski‐Păunescu).

Let

F:(\mathbb{R}^{d}\mathrm{x}I,\{0\}\times I)\rightarrow(\mathbb{R},0)

beaNash germ. Then the

germs

[t(X)=F(t,x):\mathrm{t}\mathbb{R}^{d},0

)\rightarrow(\mathbb{R},0),t\in I,

define locallyfinitely

many

arc‐analytic

classes.

2

A

motivic invariant

of the

arc‐analytic equivalence

Thissection isdevotedtotheinvariantof the

arc‐analytic

equivalence

introducedin

[6].

This

invariant isconstructed

following

themotiviczetafunction of Denef‐Loeser

[7]

but withco‐

efficientsinareal

analogue

of the Grothendieck

ring

introduced

by

Guibert‐Loeser‐Merle

[13].

It

generalizes

themotivic zeta functions of Koike‐Parusiriski

[14]

and of G. Fichou

[8, 9].

\starANash functionis

arealanalyticfunction withsemialgebraic graph

$\dagger$ A NashmodificationisapropersurjectiveNashmap whosecomplexificationisproper andbimeromorphic.

$\ddagger$K.Kurdyka[16,Théorème5.2]provedthata

semialgebraic arc‐analyticmapisrealanalyticoutsideasetof

(3)

Jean‐Baptiste

Campesato

Definition2.1

([20, §4.2]).

An\mathcal{A}S‐setisa

semialgebraic

subset

A\subset \mathbb{P}_{\mathrm{N}}^{n}

such that

given

areal

analytic

arc

$\gamma$:(-1,1)\rightarrow \mathbb{P}_{\mathrm{N}}^{n} satisfying

$\gamma$(-1,0)\subset A

thereexists $\epsilon$>0such that

$\gamma$(0, $\epsilon$)\subset A.

Remark2.2

([20,

§4.2]).

The \mathcal{A}S‐subsets of

\mathbb{P}_{\mathrm{R}}^{n}

form the boolean

algebra spanned

by

semi‐

algebraic

arc‐symmetric (in

thesenseofK.

Kurdyka

[16])

subsets of

\mathbb{P}_{\mathrm{R}}^{n}

.

Particularly,

\mathcal{A}Sis

stable

by

\mathrm{u},\mathrm{n},\backslash .

Definition2.3. We denote

by

K_{0}(\mathcal{A}S)

the free abelian group

spanned by symbols

[A], A\in

\mathcal{A}S modulo:

(i)

If thereisa

bijection

A\rightarrow Bwith

AS‐graph

then[A]=[B].

(ii)

IfB isaclosed AS‐subset ofAthen[A]=[A\backslash B]+[B].

Moreover,

K_{0}(AS)hasa

ring

structureinduced

by

thecartesian

product:

(iii)

[A\times B]=[A][B].

Wedenote

by

0=[\emptyset]the class of the

empty

setwhichistheunitof theaddition,

by

1=[\{*\}]

the class of the

point

whichistheunitof the

product

and

by

\mathrm{L}_{AS}=[\mathbb{R}]

the class of the affine line.

Notation 2.4. Wedenote

by

\mathcal{M}_{AS}=K_{0}\mathrm{t}AS

)

[\mathrm{L}_{A\mathcal{S}}^{-1}]

the localization of

K_{0}(AS)

with

resped

to

\{\mathrm{L}_{A\mathcal{S}}^{i}, i\in \mathrm{N}\}.

Theinterestof

working

with AS‐sets hereistheexistenceof the virtual Poincaré

poly‐

nomial.

Theorem2.5

([17][8][18]).

Thereexistsa

unique

ring morphism

$\beta$:K_{0}(AS\rangle\rightarrow \mathrm{Z}[u]

,called the virtual Poincaré

polynomial,

such

that,

if

A\in ASis

compact

and

non‐singular

then

$\beta$([A])=

$\Sigma$_{i}\dim H_{i}(A,\mathrm{Z}_{2})u^{ $\iota$}.

Moreover,the virtual Poincaré

polynomial

encodes the dimensionsince,

if

A\in ASis

nonempty,

\deg $\beta$([A])=

djmA(andthe

leading

coefficient

is

positive).

Remark2.6

([22]).

Notice that ifweomitthe

arc‐symmetric

conditiontowork with allsemi‐

algebraic

setsthenwemay deduce from the cell

decomposition

that every additiveinvariant

of the

semialgebraic

setsupto

semialgebraic homeomorphism

factorises

through

the Euler characteristic with

compact support.

In thissituation,it is

impossible

torecoverthe dimen‐

sion, since,for

example,

$\chi$_{\mathrm{c}}(S^{1})=0

(whereas

S^{1}is

nonempty).

Notice also that foran\mathcal{A}S‐set

A,

$\beta$([A])(u=-1\rangle=$\chi$_{c}(A\rangle.

Definition2.7. Wedenote

by

K_{0}(A\mathcal{S}_{\mathrm{R}}*)

the free abelian group

spanned by symbols

[$\varphi$_{X}

:

X\rightarrow \mathbb{R}^{*}],whereXandthe

graph $\Gamma$_{ $\varphi$ x}

arein\mathcal{A}S,modulo the relations:

(i)

If thereisa

bijection

h:X\rightarrow \mathrm{Y}with

AS‐graph

such that

$\varphi$_{X}=$\varphi$_{\mathrm{Y}}\circ h

then

[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]=[$\varphi$_{\mathrm{Y}}:\mathrm{Y}\rightarrow \mathbb{N}^{*}]

(ii)

If\mathrm{Y}\subset X isaclosed A\mathcal{S}‐subset then

[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]=[$\varphi$_{X|X\backslash \mathrm{Y}}:X\backslash \mathrm{Y}\rightarrow \mathbb{R}^{*}]+[$\varphi$_{X|Y}:\mathrm{Y}\rightarrow \mathbb{R}^{*}]

Thefiber

product

inducesa

ring

structure

by adding

the relation:

(iii)

[X \mathrm{x}_{\mathbb{R}}\cdot Y\rightarrow \mathbb{R}^{*}]=[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}][ $\varphi$ \mathrm{y}:\mathrm{Y}\rightarrow \mathbb{R}^{*}]

Thecartesian

product

inducesaK_{0}(AS)

‐algebra

structure

by adding

the relation:

(iv) [A][$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]=[$\varphi$_{X}\circ \mathrm{p}\mathrm{r}_{2}:A\times X\rightarrow \mathbb{R}^{*}]

Wedenote

by

0=[\emptyset] the class of the

empty

setwhichistheunitof the

addition,

by

\mathrm{T}=[\mathrm{i}\mathrm{d}:\mathbb{R}^{*}\rightarrow \mathbb{R}^{*}]

the class of the

identity

whichistheunitof the

product

and

by

\mathrm{L}=\mathrm{L}_{\mathcal{A}S}1=[\mathrm{p}\mathrm{r}_{2}:\mathbb{R}\mathrm{x}\mathbb{R}^{*}\rightarrow \mathbb{N}^{*}]

(4)

Remark2.8. The group consideredin

[6]

is

equivariant

since it isassumedthatX is

equipped

withanactionof \mathbb{R}^{*}

compatible

with$\varphi$_{X} insome sense. We alsowork with

equivariant

iso‐

morphism

classes and thusit is necessarytoadd technical relationsinorderto

identify

someclasses.

This

equivariant aspect

isomittedinthisnote to

simplify

the

presentation.

Howeverit is

necessarytoprove that the convolution formula

of[6]

is

compatible

with theone

of[14].

We

also believe thatit isneeded forabetter

comprehension

of the so‐called realmotivicMilnor

fiber.

Notation2.9. Weset

$\Lambda$ 4=K_{0}\mathrm{t}AS_{\mathbb{R}}\cdot

)

[\mathrm{L}^{-1}]

. Notice that\mathcal{M} hasanaturalstructureof

$\lambda$ 4_{AS^{-}}

algebra.

Proposition

2.10

([6, §3

Thereexistsa

unique

morphism

:

\mathcal{M}\rightarrow \mathcal{M}_{\mathcal{A}S}

of

\mathcal{M}_{AS}

‐modules

inducedon

symbols by

[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]\mapsto[X]

It iscalled the

forgetful morphism.

Proposition

2.11

([6, Proposition

4.16]).

For $\epsilon$\in\{+,-\},thereexistsa

unique

morphism

F^{ $\epsilon$}:\mathcal{M}\rightarrow

\mathcal{M}_{AS}

of

\mathcal{M}_{AS}

‐algebras

inducedon

symbols by

[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]\mapsto[$\varphi$_{X}^{-1}( $\epsilon$ 1)]

Remark2.12. The

forgetful morphism

is not

compatible

with the

ring

structures sincethe

one on\mathcal{M}isinduced

by

the fiber

product

whereas theone on

\mathcal{M}_{AS}

isinduced

by

cartesian

product.

Thisis

highlighted by

computing

$\beta$\cap \mathrm{t}=u+1\neq 1= $\beta$(1)

.

However,

the

morphisms

F^{ $\epsilon$}are

compatible

with the

ring

structures sincethe fiber

prod‐

uctover one

point

coincides with thecartesian

product.

Definition2.13. Let

f:1\mathbb{R}^{d},0

)\rightarrow(\mathbb{R},0)beaNash function germ.Wedefine the localmotivic

zetafunction of

f

by

Z_{f}(T)=\displaystyle \sum_{n\geq 1}[\mathrm{a}\mathrm{c}_{ $\gamma$}^{n}:X_{n}(f)\rightarrow \mathbb{R}^{*}]\mathrm{L}^{-nd}T^{n}\in \mathcal{M}[T1

where

X_{n}(f)=\{ $\gamma$=a_{1}t+\ldots+a_{n}t^{n}, ai\in \mathbb{R}^{d}, f( $\gamma$(t))=ct^{n}+ , c\neq 0\}

and

\mathrm{a}\mathrm{c}_{f}^{n}

:

X_{n}(f)\rightarrow \mathbb{R}^{*}

isthe

angular

component

map defined

by

\mathrm{a}\mathrm{c}_{ $\gamma$}^{n}( $\gamma$)=\mathrm{a}\mathrm{c}1f\circ $\gamma$

):=c.

Theorem 2.14

([6,

Theorem

7.11]).

If

f,g:\mathrm{t}\mathbb{R}^{d},0

)\rightarrow(\mathbb{R},0)aretwo

arc‐analytically equivalent

Nashfunction

germs then

Z_{f}(T)=Z_{g}(T)

.

The heuristic idea of the

proof

is the

following.

First,lets bea formal variable and

set T=\mathrm{L}^{-s}. Then,aftersomesmall

changes,

Z $\gamma$ \mathrm{t}T

) may beseenas amotivic

integral

with

parameter

s,whateveritmeans:

z_{f^{(T)=\int_{L(\mathrm{R}^{d},0)}\mathrm{L}^{-\mathrm{o}\mathrm{r}\mathrm{d}_{t}f\cdot s}}}

Nowassumethat

f

andgare

arc‐analytically equivalent,

then thereexists $\varphi$asinDef‐

imition 1.1.

By

a result of Bierstone‐Milman

[2]

and A. Parusmski

[19],

there exists $\sigma$ :

(5)

Jean‐Baptiste

Campesato

that\overline{ $\sigma$}= $\varphi$\circ $\sigma$isNash. Thereforewehave the

following

commutative

diagram

By

themotivic

change

of variables

formula,

we

get

Z_{f}(T)=\displaystyle \int_{\mathcal{L}(\mathrm{R}}

Ỉo

)^{\mathrm{L}^{-\mathrm{o}\mathrm{r}\mathrm{d}_{t}r\cdot s}=\int_{\mathcal{L}(M,E)}\mathrm{L}^{-\mathrm{o}\mathrm{r}\mathrm{d}_{t}(f\circ $\sigma$)\cdot s-\mathrm{o}\mathrm{r}\mathrm{d}_{t}\mathrm{J}\mathrm{a}\mathrm{c}_{ $\sigma$}}}

Sincethe

previous

diagram

commutes,

f\circ $\sigma$=g\circ

and,

by

l.l.(iii),

\mathrm{o}\mathrm{r}\mathrm{d}_{t}\mathrm{J}\mathrm{a}\mathrm{c}_{ $\sigma$}=\mathrm{o}\mathrm{r}\mathrm{d}_{t}\mathrm{J}\mathrm{a}\mathrm{c}_{\overline{ $\sigma$}}.

Then, again

by

the

change

of variables

formula,

wemay conclude

z_{t^{(T)=\int_{L(M,E)}\mathrm{L}^{-\mathrm{o}\mathrm{r}\mathrm{d}_{t}(f\circ $\sigma$)\cdot s-\mathrm{o}\mathrm{r}\mathrm{d}_{t}\mathrm{J}\mathrm{a}\mathrm{c}_{ $\sigma$}}=\int_{L(M,E)}\mathrm{L}^{-\mathrm{o}\mathrm{r}\mathrm{d}_{t}(g\circ\overline{ $\sigma$})\cdot s-\mathrm{o}\mathrm{r}\mathrm{d}_{t}\mathrm{J}\mathrm{a}\mathrm{c}_{\overline{ $\sigma$}}}=\int_{\mathcal{L}(\mathrm{R}^{d},0)}\mathrm{L}^{-\mathrm{o}\mathrm{r}\mathrm{d}_{t}g\cdot s}=Z_{g}(T)}}

Notice

that,

in

[6] (and

beforein

[14]

and

[8]),

weavoidtointroduce themotivicmeasure

(for

whichwewould needtowork witha

completion

of \mathcal{M}

)

and themotivic

integral.

For this purpose, the

change

of variable formulaishiddenina

computation

of

Z_{f}(T)

in terms

of $\sigma$

directly

with the coefficientsof

Z_{f}(T)

as apowerseries in T,inaway similartoDenef‐

Loeserfor their

proof

of the

rationality

of theirmotivic zetafunctions. Thenwecompare

these rational formulae of

Z_{f}(T)

and

Z_{g}(T)

toconclude.

3

A

convolution formula

Proposition

3.1.Thereexistsa

unique K_{0}(AS)

‐bilinear

map*:K_{0}(\mathcal{A}S_{\mathrm{R}^{n}})\mathrm{x}K_{0}\mathrm{t}\mathcal{A}S_{\mathbb{R}}*

)

\rightarrow K_{0}(\mathcal{A}S_{\mathbb{R}}*)

satisfying

the

following

relationon

symbols

[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]*[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]

=-[$\varphi$_{X}+$\varphi$_{\mathrm{Y}}:X\mathrm{x}\mathrm{Y}\backslash ($\varphi$_{X}+ $\varphi$ \mathrm{y})^{-1}(0)\rightarrow \mathbb{R}^{*}]+[\mathrm{p}\mathrm{r}_{2}:\mathrm{t}$\varphi$_{X}+$\varphi$_{\mathrm{Y}})^{-1}(0)\times \mathbb{R}^{*}\rightarrow \mathbb{R}^{*}]

It iscalled the convolution

product.

Remark3.2. It inducesa

\mathcal{M}_{AS}

‐bilinear \mathrm{m}\mathrm{a}\mathrm{p}*: \mathcal{M}\times \mathcal{M}\rightarrow 4. Itisassociative,commutative

anditadmits1asunit.

Definition3.3. lhe modifiedzetafunction ofaNash function germ

f

:

(\mathbb{R}^{d},0)\rightarrow(\mathbb{R},0)

is

defined

by

\tilde{z}_{f^{(T)}}=z_{r^{(T)-\frac{1-Z_{f}^{\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}(T)}{\mathrm{T}-T}}}+\mathrm{T}

where

Z_{f}^{\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}(T)

isdefined

by

applying

$\alpha$\mapsto\overline{a}1coefficientwiseto

z_{r}(T)

.

Remark3.4

([6,

Corollary

6.14]).

The modifiedzetafunction and thezetafunction encode thesameinformationsince

(6)

Theorem3.5

(The

convolution formula

[6,

Theorem

6.15]).

Fori=1,2,let

f_{i}:(\mathbb{R}^{d_{i}},0)\rightarrow \mathrm{t}\mathbb{R},0

) hea

Nashfunction

germ and

define f_{1}\oplus h:(\mathbb{R}^{d_{1}}\mathrm{x}\mathbb{R}^{d_{2}},0)\rightarrow(\mathbb{R},0)

by

f_{1}\oplus f_{2}(x_{1},x2)=f_{1}(x_{1})+f_{2}\mathrm{t}x2

).

Then

\overline{z}_{f_{1}\oplus h^{(T)=-\overline{Z}_{f_{1}}\mathrm{t}T\rangle \mathrm{O}\tilde{Z}$\gamma$_{2}\mathrm{t}T)}}

where\mathrm{O} is

defined

by applying

the convolution

product

*

coefficientwise.

The idea of the

proof

is the

following.

Assume thatwewantto

compute X_{n}(f_{1}\oplus f_{2}),

i.e. welook for

$\gamma$_{1}(t)

and$\gamma$_{2}( t\ranglesuch that

f_{1}($\gamma$_{1}\mathrm{t}t)

)

+f_{2}($\gamma$_{2}(t))=ct^{n}+

c\neq 0. Assume that

f_{1}\mathrm{t}$\gamma$_{1}(t))=c_{1}t^{n}1+

and

f_{2}\mathrm{t}$\gamma$_{2}(t\rangle)=c_{2}t^{n}2+

Weencounterthe

following

cases:

1. n1=n2=nandc_{1}+c_{2}\neq 0,inthiscasec=c_{1}+c2.

2. n_{1}=n2<nand c_{1}+c_{2}=0.

3. n_{1}=n<n2,inthiscasec=c_{1}.

4. n_{1}>n_{2}=n,inthiscasec=c_{2}.

Thetwofirstitemsare

naturally

handled

by

the definition of the convolution

product.

The

twolastitemsare

why

weneed towork with the modified zeta

function).

For technical

reasons, inthecurrent

proof,

weneedtowork witharesolution off_{i} inordertodo the

required

computations.

4

Applications:

some

classification results

4.1

Arc‐analytic

classification of

Brieskorn

polynomials

Definition4.1. A

polynomial

f\in \mathbb{R}[x_{1},\ldots,xd]

issaidtobeaBrieskorn

polynomial

ifit isof the

following

form

f(x)=\displaystyle \sum_{i=1}^{cl}$\epsilon$_{i}x_{i}^{h_{i}}, $\epsilon$_{i}\neq 0, h_{i}\geq 1

Sincewe are

only

interestedinthe

arc‐analytic

classification of Brieskorn

polynomials,

wefirst do the

following simplifications.

Remark4.2. Sincewemay reorder the variables without

changing

the

arc‐analytic

type

of a

polynomial,

wewill

always

assumethat

h_{1}\leq k_{2}\leq \leq k_{d}

Inthesame

vein,.

wemayassumethat $\epsilon$ i=\pm 1.

Remark 4.3. Wemay first elude the

non‐singular

case.

Indeed,

aBrieskorn

polynomial

f(x)=$\Sigma$_{i=1}^{d}$\epsilon$_{i}x_{i}^{h_{i}}

is

non‐singular

if and

only

if thereexistsi=1 ,dsuch that

h_{i}=1

.Without

loss of

generality,

wemayassumeinthiscasethath_{1}=1.

Then,

fis

arc‐analytically

equiv‐

alentto (xl,\cdots, xd)\mapsto X1

by applying

the Nashinverse

mapping

theorem to(xl,\cdots, xd)\mapsto

(f(x),x_{2} ,xd)

. Notice

that,

inthiscase,

\overline{Z}_{f}(T)=0.

Fromnowon,we assumethatk_{i}\geq 2.

The

following

theoremis areal

analogue

ofaresultof

Yoshinaga‐Suzuki

[26] stating

that the

topological

type

ofaBrieskorn

singularity

determinesits

exponents.

Theorem4.4

([6,

Corollary 8.41).

Assumethat the Brieskorn

polynomials

f(x)=\displaystyle \sum_{i=1}^{d}$\epsilon$_{i}x_{i}^{h_{i}}

and

g(x)=\displaystyle \sum_{i=1}^{d}$\eta$_{i}x_{i}^{l_{i}}

with

(7)

Jean‐Baptiste

Campesato

are

arc‐analytically

equivalent,

then

\forall i=1,\ldots,d, k_{i}=l_{i}

Since the modifiedmotiviczetafunctionisaninvariantof the

arc‐analytic

equivalence,

it is

enough

toshow thatwemayrecoverthe

exponents

ofaBrieskorn

polynomial

f

from

\overline{Z}_{f}\mathrm{t}T)

.This fact may be

proved following

thenext

plan

dividedinthree

steps.

1. First,

by

theconvolutionformula,wemay deduce the modifiedzetafunction

\tilde{Z}_{f}(T)

of

f

from theone

\tilde{Z}_{ $\epsilon$ x^{h}}

ofapure monomial

$\epsilon$ x^{k}

. An easy

computation gives

\tilde{Z}_{ $\epsilon$ x^{k}}(T)=-T-\cdots-T^{h-1}

-(1-[ $\epsilon$ x^{k}:\mathbb{R}^{*}\rightarrow \mathbb{R}^{*}])\mathrm{L}^{-1}T^{h}-\mathrm{L}^{-1}T^{k+1} -\mathrm{L}^{-1}T^{2h-1}

-(\mathrm{t}-[ $\epsilon$ x^{k}:\mathbb{R}^{*}\rightarrow \mathbb{R}^{*}])\mathrm{L}^{-2}T^{2h}-\mathrm{L}^{-2}T^{2h+1}-\ldots-\mathrm{L}^{-2}T^{3h-1}

Particularly,

by

the convolution

formula,

ifnis nota

multiple

ofan

exponent

h_{i},the

coefficienta_{n}ofT^{n} in

\overline{Z}_{f}(T\rangle

is

-\mathrm{L}^{-$\Sigma$_{i=1}^{d}\lfloor_{$\Gamma$_{i}}^{n}\rfloor}.

2. Next,wededucefrom thisanupper bound of

k_{d}

.

Indeed,

ifpisa

prime

number

big

enough,

pis nota

multiple

ofan

exponent

k_{i},then

\displaystyle \lim_{p\mathrm{p}\dot{\mathrm{n}}\mathrm{m}\mathrm{e}}\frac{1-\deg $\beta$(\overline{a_{p}})}{p}=\lim_{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}}\frac{$\Sigma$_{i=1}^{d}\lfloor\not\in_{i}\rfloor}{p}=\sum_{i=1}^{d}\frac{1}{h_{i}}

Sincethereare

only finitely

many

(hí,

\cdots,

k_{d}'

) such that

$\Sigma$_{i=1}^{d}\displaystyle \frac{1}{k_{i}}=$\Sigma$_{i=1h_{i}}^{d1}\neg

,wemay

de‐

duce from

\overline{Z}_{f}\mathrm{t}T

)anupper boundKof

\{h_{1},\ldots,k_{d}\}.

3. Weconclude

by

constructing

from the coefficients of

\tilde{Z}_{f}(T)

alinear

system

intheun‐

knowns

M_{k}=\#\{i, k_{i}=h\}

,for h=1 ,K,andwesolveit.

Sincewe are

working

with real

numbers,

it isnaturaltowonder whatisthe

impact

of the

signs

of the coefficients oÍfonits

arc‐analytic

type.

Some

preliminary

resultswe

present

below allowus to statethe

following

conjecture

telling

that themotivic zetafunctionis

a

complete

invariantof the

arc‐analytic

type

ofaBrieskorn

polynomial.

They

also

give

conditionsonthe

exponents

and coefficients ofaBrieskorn

polynomial

tocharacterizeits

arc‐analytic

type.

Conjecture

4.5

([3,

Conjecture

1.10.1]).

Let

f\displaystyle \mathrm{t}x)=\sum_{i=1}^{d}$\epsilon$_{i}x_{i}^{k_{i}}

and

g(x)=\displaystyle \sum_{i=1}^{d}$\eta$_{i}x_{i}^{t_{i}}

betwoBrieskorn

polynomials

with

$\epsilon$_{i},$\eta$_{i}\in\{\pm 1\}

. Weassumethat 2\leq h_{1}\leq \leq k_{d}and 2\leq l_{1}\leq \leq

l_{d},

and,

moreover,that

if

k_{i}=h_{i+1}= =k_{i+m}then$\epsilon$_{i}\geq \geq$\epsilon$_{i+m}

(resp. if

l_{i}=l_{i+1}= =l_{i+m} then $\eta$_{i}\geq \geq$\eta$_{i+m}).

Then the

following

are

equivalent:

(I) fandgare

arc‐analytically

equivalent.

(2)

z_{f^{(T)=Z_{g}(T)}}

(3) (i) \forall i,k_{i}=t_{i}

(8)

First,noticethatthis

conjecture

is

compatible

with the classifications of Koike‐Parusmski

inthetwovariablecaseandof G. Fichouinthe three variablecase.

We have

already

shown that

4.5.(1)\Rightarrow 4.5.(2)

and that

4.5.(2)\Rightarrow 4.5.(3).(\mathrm{i})

. It is

already

known

[3,

Lemme

1.10.2]

that

4.5.(3)\Rightarrow 4.5.(1)

. The ideatoprove this last

step

is to

adapt

an

argument

ofKoike‐Parusiński

[14,

p2095]

which consists in

embedding

[

andg in\mathrm{a}

same

family

of Nash function germs with isolated

singularities

andtousethe absence of

continuousmodulitoconclude.

We end thissection

by

giving,

whichwebelievetobe, a

promising

waytoprove the

previous conjecture.

Our

goal

is toprove

4.5.(2)\Rightarrow 4.5.(3).(\mathrm{i}\mathrm{i})

.

Again,

let

f

bea Urieskorn

polynomial

and definea_{n}

by

\tilde{Z}_{f}1T

)

=$\Sigma$_{n\geq 1}a_{n}T^{n}

. Assume that n\geq 1 isa

multiple

ofan

even

exponent

of

f

butis nota

multiple

ofanodd

exponent. Then,

by

acloser lookatthe

convolution

\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}_{J}

we

get

that

$\beta$(\displaystyle \overline{a_{n}})u^{$\Sigma$_{i=1}^{d}\lfloor\frac{n}{k_{i}}\rfloor}= $\beta$(\sum_{i,k_{i}|n}$\epsilon$_{i}x_{i}^{k_{i}}\neq 0)-(u-1) $\beta$(\sum_{i,h_{i}|n}$\epsilon$_{i}x_{i}^{h_{i}}=0)=u^{t/\mathrm{t}i,h_{i}|n\}}-u $\beta$(\sum_{i,k_{i}|n}$\epsilon$_{i}x_{i}^{h_{i}}=0)

$\beta$(F^{ $\epsilon$}\displaystyle \mathrm{t}a_{n}))u^{$\Sigma$_{i=1}^{d}\lfloor\frac{n}{h_{i}}\rfloor}= $\beta$(\sum_{i,k_{i}|n}$\epsilon$_{i}x_{i}^{h_{i}}= $\epsilon$ 1)- $\beta$(\sum_{i,h_{i}|n}$\epsilon$_{i}x_{i}^{h_{i}}=0)

Sothat,for $\epsilon$=+ wemayrecover

$\beta$($\Sigma$_{i,k_{i}|n^{\mathcal{E}}i}x_{i}^{k_{i}}= $\epsilon$ 1)

from

\tilde{Z}_{f}(T)

:

$\beta$(i

Hence, ifwe were able to find the number of

positive (or

negative)

coefficients ofa

Brieskorn

polynomial

with even

exponents

from the virtual Poincare

polynomials

ofits

preimages

over1 and -1,wecould conclude

by

induction. We believe thatit is

possible

sothat the

conjecture

isreducedto the

computation

ofsomevirtual Poincaré

polynomi‐

als. Noticeit

already

holds fora

homogeneous

Brieskorn

polynomial

ofeven

degree

by

[10,

Corollary

2.5 &

Corollary

2.6].

4.2

Arc‐analytic

classification of some

weighted

homogeneous

polyno‐

mials

In the

complex

case, it isknown that the local

analytic

type

ofa

singular

weighted

homoge

neous

polynomial

with isolated

singularity

atthe

origin

determinesits

weights

[24].

Italso

holds

by

considering merely

the

topological

type

in two

[27]

and three

[23]

variables.

T.Fukui

[12,

Conjecture

9.2] conjectures

the real

counterpart

for

weighted homogeneous

real

polynomials

with isolated

singularity

inthe

blow‐analytic

context.This

conjecture

has been proven

by

O. M. Abderrahmane

[1]

in twovariables and

by

G.Fichou andT.Fukui

[11]

inthree variablesinthe blow‐Nashcontextforconvenient

weighted homogeneous polyno‐

mials whichare

non‐degenerate

with

respect

totheir Newton

polyhedra.

Since Brieskorn

polynomials

areconvenient

weighted

homogeneous polynomials

which

are

non‐degenerate

with

respect

totheir Newton

polyhedra,

it isnaturaltoask whether the material

presented

inthe

previous

sectionwould allowoneto

generalize

the result of G. Fichou andT.Fukui withnoconditiononthe number of variables.

Afirst obstacleisthatwecan’tusethe convolution formula anymoresincewecan’tas‐

sumethat sucha

polynomial

isasumof pure monomials. However, it isstill

possible

to

adapt

the

strategy

usedtoprove that the

arc‐analytic

type

ofaBrieskorn

polynomial

de‐

termines its

exponents.

Itrelieson aformulato

compute

the modifiedzetafunction ofa

(9)

Jean‐Baptiste Campesato

Theorem

4.6([5]).

Let

f,g\in \mathbb{R}[x_{1},\ldots,xd]

betwo

arc‐analytically

equivalent weighted homogeneous

polynomials

whichare

non‐degenerate

with

respect

totheir Newton

polyhedra.

Then

1. Either

they

areboth

non‐singular,

andinthiscase

they

areboth

arc‐analytically

equivalent

to

(xl,\cdots x )

2. Or

they

share thesame

weights(up

to

permutation

and

positive

common

multiplicativefactor).

Acknowledgements.

The authoris

supported

by

a

Japan Society

for the Promotion of Sci‐

ence

(JSPS)

Postdoctoral

Fellowship

(Short‐term)

for North American and

European

Re‐

searchers.

References

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[2] E. \mathrm{B}\mathrm{I}\mathrm{E}\mathrm{R}\mathfrak{N}\mathrm{o}\mathrm{N}\mathrm{E}ANDP. D.MILMAN,Arc‐analyticfunctions,Invent.Math..101(1990),pp.411‐424.

[3] J.‐B.CAMPESATO,Unefonctionzêtamotiviquepour l’étude dessingularitisrédles,PhDthesis,Umversité Nice

Sophia Antipolis,122015.

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194.

[5] —,Onthearc‐analytictypeofsome weighted homogeneous polynomials,2016,aJXiV:1612.08269.

[6] —,Onamotivicinvariantofthe arc‐analytic equivalence,Ann. Inst.Fourier(Grenoble),67(2017)_{J}pp. 143‐196.

[7] }.DENEFAND $\Gamma$.LOESER,MotivicIgusazetafunctions,}.AlgebraicGeom.,7(1998),pp. 505‐537.

[8] G.FICHOU,Motivic invariantsofarc‐symmetricsetsandblow‐Nashequivalence,Compos.Math., 141 (2005),

pp. 655‐688.

[9] —,Zetafunctionsand blow‐Nashequivalence,Ann.Polon.Math.,87(2005),pp.111‐126.

[IO] —,The corankand the indexareblow‐Nash inoniants,Kodai Math.J.,29(2006),pp. 3140.

[11] G. FICHOUANDT.FUKUI,Motivic invariantsofrealpolynomialfunctionsand their Newtonpolyhedrons,Math. Proc.

CambridgePhilos.Soc.,160(2016),pp.141‐166.

[12] T.FUKUi,Seekinginvariantsfor blow‐analytic equivalence,Compositio Math.,105(1997),pp. 95‐108.

[13] G.GUiBEPJ, $\Gamma$.IOESER,ANDM.MERLE,lteratedvanishing cycles,convolution,andamotivicanalogue ofaconjecture

ofSteenbrink,Duke Math.I.,132(2006),pp. 409\ovalbox{\tt\small REJECT} 457.

[14] S. KOIKEANDA.PARUSiN’SKJ,Motivic‐typeinvariantsof blơw‐analytic equivalence,Ann.Inst. Fourier(Grenoble),

53(2003),pp. 2061‐2104.

[15] T.‐C.Kuo,Onclassification ofrealsingularities,Invent.Math.,82(1985),pp.257‐262.

[16] K.KURDYKA,Ensemblessemi‐algébriquessymétriquespararcs,Math.Ann.,282(1988),pp. 445A62.

[17] C. MCCRORYANDA.PARusiNsKi,VirtualBettinumbersofrealalgebraicvarieties,C.R.Math. Acad. Sci.Paris,336

(2003),pp. 763‐768.

[18] —,Thewdghifiltrationforrealalgebraicvarieties,inTopologyof stratified spaces,vol. 58ofMath.Sci. Res.

Inst.Publ.,CambridgeUniv.Press,Cambridge,2011,pp.121‐160.

[19] A\mathrm{P}\mathrm{A}\mathrm{R}\mathrm{U}\mathrm{s}\mathrm{I}\mathrm{N}'\mathrm{S}\mathrm{K}\mathrm{i},Subanalyticfunctions,Trans.Amer. Math.Soc.,344(1994),pp. 583‐595.

[20] —,Topology ofinjectiveendomorphisms ofrealalgebraicsets, Math.Ann.,328(2004),pp. 353‐372.

[21] A.\mathrm{P}_{\mathrm{A}\mathrm{R}}\mathrm{u}\mathrm{s}\mathrm{i}'NSKiANDL.PĂUNESCU,Arc‐wiseanalyticstratitication, Whitneyfibenngconjectureand Zariskiequisingu‐

lanty,Adv.Math.,309(2017),pp. 254‐305.

[22] R.QUAREZ, Espacedes germes d’arcs réelsetséne dePoincaréd’un ensemblesemi‐algéUrique,Ann. hs8. Fourier

(Grenoble),51(20m),pp. 4$48.

[23] O.SAEKI,Topologicalinvarianceofweightsfor weighted homogeneousisolatedsingularitiesin\mathrm{C}^{3},Proc. Amer. Math.

Soc.,103(1988),pp. 905‐909.

[24] K.SAITO,QuasihomogeneisolierteSingularitatenvonHyperflächen,Invent.Math.,14(1971),pp. 123‐142.

[25] H.WHITNEY,Localpropertiesqfanalyticvarieties,inDifferential and CombinatorialTopology(A Symposiumin

Honor of MarstonMorse),Princeton Uiuv.Press, Prmceton,N.J.,1965,pp.205‐244.

[26] E.YOSHINAGAANDM.SUZUKI,On thetopologicaltypesofsingularities ofBrieskorn‐Phamtype,SctRep.Yokohama

(10)

[27] —,Topologicaltypesofquasi7iomogeneous singularitiesin\mathbb{C}^{2},Topology,18(1979),pp.113‐116.

Jean‐Baptiste Campesato

DepartmentoỉMathematics,FacultyofScience,

SaitamaUniversity,

255Shimo‐Okubo,Sakura‐ku,

Saitama338‐8570,

Japan.

参照

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