On
someclassification 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
versionofatalkgivenduring
the conferenceSingularity
theory ofdifferential
maps andifsapplications
attheRIMS,Kyoto
(December6‐9,2016).We firststatethe definition andsomepropertiesof the
arc‐analytic equivalence
whichisan
equivalence
relation withnocontinuousmodulionNash(i.e.realanalytic
andsemi‐algebraic)
function germs. Itisasemiaigebraic
versionof theblow‐analytic equivalence
of T.‐C. Kuo.
Then,wepresentaninvariantof the
arc‐analytic equivalence
whichisconstructedfollowing
themotivic zetafunction of Denef‐Loeser.Finally,
weexplain
howtoderive fromitsomeclassification results for Brieskompoly‐
nomialsandmore
generally
forsomeweighted homogeneous
polynomials.
Contents
1 The
arc‐analytic
equivalence
12 A motivic invariantof the
arc‐analytic equivalence
23 Aconvolution formula 5
4
Applications:
someclassification \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\{\mathrm{s} ó4.1
Arc‐analytic
classification of Brieskornpolynomials
. . . 64.2
Arc‐analytic
classification ofsomeweighted homogeneous polynomials
. . . 8References 9
1
The
\mathrm{a}\mathrm{r}\mathrm{c}\leftrightarrowanalytic equivalence
H.
Whitney
[25,
Example
13.1]
noticed that thecross‐ratio isacontinuousmodulusof thefamily f_{t}:(\mathbb{R}^{2},0)\rightarrow \mathrm{t}\mathbb{R},0
), t\in \mathrm{t}0,1), definedby
f_{t}(x,y)=xy(y-x)(y-tx).Particularly,
twodistinct function germs of this
family
are neverC^{1}
‐equivalent,
i.e. if thereexists a C^{1_{-}}diffeomorphism
$\varphi$:(\mathbb{R}^{2},0)\rightarrow(\mathbb{R}^{2},0)
such thatf_{t^{l}}=f_{t}\circ $\varphi$
then t=t'.T.‐C. Kuo
[15]
suggested
theblow‐analytic equivalence
as acandidatetoobtainaclas‐sification of real
singularities
withoutcontinuousmoduli.Heproved
that thisnotion isanequivalence
relationonrealanalytic
function germs and thatitadmitsnocontinuousmod‐uli for isolated
singularities.
Indeed,
afamily
of realanalytic
function germs with isolatedsingularities
defineslocally finitely
manyblow‐analytic equivalence
classes.Up
tonow,the knowninvariantsof theblow‐analytic equivalence
arethe Fukuiinvari‐ants
[12]
and the Koike‐Parusinskizetafunctions[14].
Inorderto constructricherinvari‐called the blow‐Nash
equivalence.
It isarelationonNash\star function germs withnocontin‐uousmoduli for isolated
singularities.
Thenotionof blow‐Nashequivalence
evolved and stabilizedtothefollowing:
twoNash function germsf,g:(\mathbb{R}^{d},0)\rightarrow(\mathbb{R},0)
areblow‐Nashequivalent
if,
afterbeing composed
with Nash modifications $\dagger$,
they
areNash‐equivalent
viaa
Nash‐diffeomorphism
which preserves themultiplicities
of theJacobian
determinants of the modifications.Initially
itwasexpected,
butnotknownyet,
whether this relationisan
equivalence
relationonNashfunction germs.The
goal
of thissection istointroducethearc‐analyt
\mathrm{c}equivalence
definedin[6].
It isacharacterization of the blow‐Nash
equivalence
in termsofarc‐analytic
maps. It avoidstoinvolve Nash modifications andit is an
equivalence
relation. Moreover,A.ParusirskiandL.Păunescu
[21]
recently proved
itadmitsnocontinuousmoduli,evenfor families ofnon‐isolated
singularities.
Definition1.1
([6,
Definition7.5]).
TwoNash function germsf,g:(\mathbb{R}^{d},0)\rightarrow \mathrm{t}\mathbb{R},0
)arearc‐analytically equivalent
if thereexistsasemialgebraic homeomorphism
$\varphi$:(\mathbb{R}^{d},0)\rightarrow(\mathbb{R}^{d},0)
such that
(i) g=f\mathrm{o} $\varphi$,
(ii)
$\varphi$ isarc‐analytic,
i.e. for$\gamma$:(\mathbb{R},0)\rightarrow(\mathbb{R}^{d},0\rangle
realanalytic,
thecomposition
$\varphi$\circ $\gamma$ isalsoreal
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 $\varphi$asintheprevious
definition,theconverse$\varphi$^{-1}
isalsoarc‐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
thefollowing
proposition.
Proposition
1.3([6,
Proposition
7.7]).
Thearc‐analytic equivalence
isanequivalence
relationonNashfunction
germs(\mathbb{R}^{d},0)\rightarrow \mathrm{t}\mathbb{R},0
).The
following
proposition
statesthat thearc‐analytic equivalence
isacharacterizationof the blow‐Nash
equivalence.
Particularly,
the blow‐Nashequivalence
isanequivalence
relationas
expected.
Proposition
1.4([6,
Proposition
7.9]).
TwoNashfunction
germsarearc‐analytically equivalent
ifand only
if
they
areblow‐Nashequivalent.
The
following
resultensuresthat thearc‐analytic
equivalence
hasnocontinuousmod‐uli,evenfor families of non‐isolated
singularities.
Itisaconsequence of[21,
Theorem8.5]
together
with theproof
of[21,
Theorem3.3]
and formula[21, (3.9)].
Theorem1.5
(Parusiski‐Păunescu).
LetF:(\mathbb{R}^{d}\mathrm{x}I,\{0\}\times I)\rightarrow(\mathbb{R},0)
beaNash germ. Then thegerms
[t(X)=F(t,x):\mathrm{t}\mathbb{R}^{d},0
)\rightarrow(\mathbb{R},0),t\in I,define locallyfinitely
manyarc‐analytic
classes.2
A
motivic invariant
of the
arc‐analytic equivalence
Thissection isdevotedtotheinvariantof the
arc‐analytic
equivalence
introducedin[6].
Thisinvariant isconstructed
following
themotiviczetafunction of Denef‐Loeser[7]
but withco‐efficientsinareal
analogue
of the Grothendieckring
introducedby
Guibert‐Loeser‐Merle[13].
Itgeneralizes
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
Jean‐Baptiste
Campesato
Definition2.1
([20, §4.2]).
An\mathcal{A}S‐setisasemialgebraic
subsetA\subset \mathbb{P}_{\mathrm{N}}^{n}
such thatgiven
arealanalytic
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 booleanalgebra spanned
by
semi‐algebraic
arc‐symmetric (in
thesenseofK.Kurdyka
[16])
subsets of\mathbb{P}_{\mathrm{R}}^{n}
.Particularly,
\mathcal{A}Sisstable
by
\mathrm{u},\mathrm{n},\backslash .Definition2.3. We denote
by
K_{0}(\mathcal{A}S)
the free abelian groupspanned by symbols
[A], A\in\mathcal{A}S modulo:
(i)
If thereisabijection
A\rightarrow BwithAS‐graph
then[A]=[B].(ii)
IfB isaclosed AS‐subset ofAthen[A]=[A\backslash B]+[B].Moreover,
K_{0}(AS)hasaring
structureinducedby
thecartesianproduct:
(iii)
[A\times B]=[A][B].Wedenote
by
0=[\emptyset]the class of theempty
setwhichistheunitof theaddition,by
1=[\{*\}]the class of the
point
whichistheunitof theproduct
andby
\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 ofK_{0}(AS)
withresped
to
\{\mathrm{L}_{A\mathcal{S}}^{i}, i\in \mathrm{N}\}.
Theinterestof
working
with AS‐sets hereistheexistenceof the virtual Poincarépoly‐
nomial.
Theorem2.5
([17][8][18]).
Thereexistsaunique
ring morphism
$\beta$:K_{0}(AS\rangle\rightarrow \mathrm{Z}[u]
,called the virtual Poincarépolynomial,
suchthat,
if
A\in ASiscompact
andnon‐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 ASisnonempty,
\deg $\beta$([A])=
djmA(andtheleading
coefficient
ispositive).
Remark2.6
([22]).
Notice that ifweomitthearc‐symmetric
conditiontowork with allsemi‐algebraic
setsthenwemay deduce from the celldecomposition
that every additiveinvariantof the
semialgebraic
setsuptosemialgebraic homeomorphism
factorisesthrough
the Euler characteristic withcompact support.
In thissituation,it isimpossible
torecoverthe dimen‐sion, since,for
example,
$\chi$_{\mathrm{c}}(S^{1})=0
(whereas
S^{1}isnonempty).
Notice also that foran\mathcal{A}S‐setA,
$\beta$([A])(u=-1\rangle=$\chi$_{c}(A\rangle.
Definition2.7. Wedenote
by
K_{0}(A\mathcal{S}_{\mathrm{R}}*)
the free abelian groupspanned by symbols
[$\varphi$_{X}
:X\rightarrow \mathbb{R}^{*}],whereXandthe
graph $\Gamma$_{ $\varphi$ x}
arein\mathcal{A}S,modulo the relations:(i)
If thereisabijection
h:X\rightarrow \mathrm{Y}withAS‐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}^{*}]
Thefiberproduct
inducesaring
structureby 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
structureby 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 theempty
setwhichistheunitof theaddition,
by
\mathrm{T}=[\mathrm{i}\mathrm{d}:\mathbb{R}^{*}\rightarrow \mathbb{R}^{*}]
the class of the
identity
whichistheunitof theproduct
andby
\mathrm{L}=\mathrm{L}_{\mathcal{A}S}1=[\mathrm{p}\mathrm{r}_{2}:\mathbb{R}\mathrm{x}\mathbb{R}^{*}\rightarrow \mathbb{N}^{*}]
Remark2.8. The group consideredin
[6]
isequivariant
since it isassumedthatX isequipped
withanactionof \mathbb{R}^{*}
compatible
with$\varphi$_{X} insome sense. We alsowork withequivariant
iso‐morphism
classes and thusit is necessarytoadd technical relationsinordertoidentify
someclasses.
This
equivariant aspect
isomittedinthisnote tosimplify
thepresentation.
Howeverit isnecessarytoprove that the convolution formula
of[6]
iscompatible
with theoneof[14].
Wealso believe thatit isneeded forabetter
comprehension
of the so‐called realmotivicMilnorfiber.
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
Thereexistsaunique
morphism
:\mathcal{M}\rightarrow \mathcal{M}_{\mathcal{A}S}
of
\mathcal{M}_{AS}
‐modulesinducedon
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\{+,-\},thereexistsaunique
morphism
F^{ $\epsilon$}:\mathcal{M}\rightarrow\mathcal{M}_{AS}
of
\mathcal{M}_{AS}
‐algebras
inducedonsymbols by
[$\varphi$_{X}:X\rightarrow \mathbb{R}^{*}]\mapsto[$\varphi$_{X}^{-1}( $\epsilon$ 1)]
Remark2.12. The
forgetful morphism
is notcompatible
with thering
structures sincetheone on\mathcal{M}isinduced
by
the fiberproduct
whereas theone on\mathcal{M}_{AS}
isinducedby
cartesianproduct.
Thisishighlighted by
computing
$\beta$\cap \mathrm{t}=u+1\neq 1= $\beta$(1)
.However,
themorphisms
F^{ $\epsilon$}arecompatible
with thering
structures sincethe fiberprod‐
uctover one
point
coincides with thecartesianproduct.
Definition2.13. Let
f:1\mathbb{R}^{d},0
)\rightarrow(\mathbb{R},0)beaNash function germ.Wedefine the localmotiviczetafunction 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}^{*}
istheangular
component
map definedby
\mathrm{a}\mathrm{c}_{ $\gamma$}^{n}( $\gamma$)=\mathrm{a}\mathrm{c}1f\circ $\gamma$
):=c.Theorem 2.14
([6,
Theorem7.11]).
If
f,g:\mathrm{t}\mathbb{R}^{d},0
)\rightarrow(\mathbb{R},0)aretwoarc‐analytically equivalent
Nashfunction
germs thenZ_{f}(T)=Z_{g}(T)
.The heuristic idea of the
proof
is thefollowing.
First,lets bea formal variable andset T=\mathrm{L}^{-s}. Then,aftersomesmall
changes,
Z $\gamma$ \mathrm{t}T
) may beseenas amotivicintegral
withparameter
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
andgarearc‐analytically equivalent,
then thereexists $\varphi$asinDef‐imition 1.1.
By
a result of Bierstone‐Milman[2]
and A. Parusmski[19],
there exists $\sigma$ :Jean‐Baptiste
Campesato
that\overline{ $\sigma$}= $\varphi$\circ $\sigma$isNash. Thereforewehave the
following
commutativediagram
By
themotivicchange
of variablesformula,
weget
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
ỡand,
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
thechange
of variablesformula,
wemay concludez_{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 withacompletion
of \mathcal{M})
and themotivicintegral.
For this purpose, thechange
of variable formulaishiddeninacomputation
ofZ_{f}(T)
in termsof $\sigma$
directly
with the coefficientsofZ_{f}(T)
as apowerseries in T,inaway similartoDenef‐Loeserfor their
proof
of therationality
of theirmotivic zetafunctions. Thenwecomparethese rational formulae of
Z_{f}(T)
andZ_{g}(T)
toconclude.3
A
convolution formula
Proposition
3.1.Thereexistsaunique K_{0}(AS)
‐bilinearmap*: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
thefollowing
relationonsymbols
[$\varphi$_{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,commutativeanditadmits1asunit.
Definition3.3. lhe modifiedzetafunction ofaNash function germ
f
:(\mathbb{R}^{d},0)\rightarrow(\mathbb{R},0)
isdefined
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)
isdefinedby
applying
$\alpha$\mapsto\overline{a}1coefficientwisetoz_{r}(T)
.Remark3.4
([6,
Corollary
6.14]).
The modifiedzetafunction and thezetafunction encode thesameinformationsinceTheorem3.5
(The
convolution formula[6,
Theorem6.15]).
Fori=1,2,letf_{i}:(\mathbb{R}^{d_{i}},0)\rightarrow \mathrm{t}\mathbb{R},0
) heaNashfunction
germ anddefine 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 convolutionproduct
*coefficientwise.
The idea of the
proof
is thefollowing.
Assume thatwewanttocompute X_{n}(f_{1}\oplus f_{2}),
i.e. welook for$\gamma$_{1}(t)
and$\gamma$_{2}( t\ranglesuch thatf_{1}($\gamma$_{1}\mathrm{t}t)
)+f_{2}($\gamma$_{2}(t))=ct^{n}+
c\neq 0. Assume thatf_{1}\mathrm{t}$\gamma$_{1}(t))=c_{1}t^{n}1+
andf_{2}\mathrm{t}$\gamma$_{2}(t\rangle)=c_{2}t^{n}2+
Weencounterthefollowing
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
handledby
the definition of the convolutionproduct.
Thetwolastitemsare
why
weneed towork with the modified zetafunction).
For technicalreasons, inthecurrent
proof,
weneedtowork witharesolution off_{i} inordertodo therequired
computations.
4
Applications:
someclassification results
4.1
Arc‐analytic
classification ofBrieskorn
polynomials
Definition4.1. A
polynomial
f\in \mathbb{R}[x_{1},\ldots,xd]
issaidtobeaBrieskornpolynomial
ifit isof thefollowing
formf(x)=\displaystyle \sum_{i=1}^{cl}$\epsilon$_{i}x_{i}^{h_{i}}, $\epsilon$_{i}\neq 0, h_{i}\geq 1
Sincewe are
only
interestedinthearc‐analytic
classification of Brieskornpolynomials,
wefirst do the
following simplifications.
Remark4.2. Sincewemay reorder the variables without
changing
thearc‐analytic
type
of apolynomial,
wewillalways
assumethath_{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,
aBrieskornpolynomial
f(x)=$\Sigma$_{i=1}^{d}$\epsilon$_{i}x_{i}^{h_{i}}
isnon‐singular
if andonly
if thereexistsi=1 ,dsuch thath_{i}=1
.Withoutloss of
generality,
wemayassumeinthiscasethath_{1}=1.Then,
fisarc‐analytically
equiv‐
alentto (xl,\cdots, xd)\mapsto X1
by applying
the Nashinversemapping
theorem to(xl,\cdots, xd)\mapsto(f(x),x_{2} ,xd)
. Noticethat,
inthiscase,\overline{Z}_{f}(T)=0.
Fromnowon,we assumethatk_{i}\geq 2.
The
following
theoremis arealanalogue
ofaresultofYoshinaga‐Suzuki
[26] stating
that the
topological
type
ofaBrieskornsingularity
determinesitsexponents.
Theorem4.4
([6,
Corollary 8.41).
Assumethat the Brieskornpolynomials
f(x)=\displaystyle \sum_{i=1}^{d}$\epsilon$_{i}x_{i}^{h_{i}}
andg(x)=\displaystyle \sum_{i=1}^{d}$\eta$_{i}x_{i}^{l_{i}}
withJean‐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 thatwemayrecovertheexponents
ofaBrieskornpolynomial
f
from\overline{Z}_{f}\mathrm{t}T)
.This fact may beproved following
thenextplan
dividedinthreesteps.
1. First,
by
theconvolutionformula,wemay deduce the modifiedzetafunction\tilde{Z}_{f}(T)
off
from theone\tilde{Z}_{ $\epsilon$ x^{h}}
ofapure monomial$\epsilon$ x^{k}
. An easycomputation 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 convolutionformula,
ifnis notamultiple
ofanexponent
h_{i},thecoefficienta_{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,
ifpisaprime
numberbig
enough,
pis notamultiple
ofanexponent
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
,wemayde‐
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)
alinearsystem
intheun‐knowns
M_{k}=\#\{i, k_{i}=h\}
,for h=1 ,K,andwesolveit.Sincewe are
working
with realnumbers,
it isnaturaltowonder whatistheimpact
of thesigns
of the coefficients oÍfonitsarc‐analytic
type.
Somepreliminary
resultswepresent
below allowus to statethe
following
conjecture
telling
that themotivic zetafunctionisa
complete
invariantof thearc‐analytic
type
ofaBrieskornpolynomial.
They
alsogive
conditionsonthe
exponents
and coefficients ofaBrieskornpolynomial
tocharacterizeitsarc‐analytic
type.
Conjecture
4.5([3,
Conjecture
1.10.1]).
Letf\displaystyle \mathrm{t}x)=\sum_{i=1}^{d}$\epsilon$_{i}x_{i}^{k_{i}}
andg(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 \leql_{d},
and,
moreover,thatif
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
areequivalent:
(I) fandgare
arc‐analytically
equivalent.
(2)
z_{f^{(T)=Z_{g}(T)}}
(3) (i) \forall i,k_{i}=t_{i}
First,noticethatthis
conjecture
iscompatible
with the classifications of Koike‐Parusmskiinthetwovariablecaseandof G. Fichouinthe three variablecase.
We have
already
shown that4.5.(1)\Rightarrow 4.5.(2)
and that4.5.(2)\Rightarrow 4.5.(3).(\mathrm{i})
. It isalready
known
[3,
Lemme1.10.2]
that4.5.(3)\Rightarrow 4.5.(1)
. The ideatoprove this laststep
is toadapt
an
argument
ofKoike‐Parusiński[14,
p2095]
which consists inembedding
[
andg in\mathrm{a}same
family
of Nash function germs with isolatedsingularities
andtousethe absence ofcontinuousmodulitoconclude.
We end thissection
by
giving,
whichwebelievetobe, apromising
waytoprove theprevious conjecture.
Ourgoal
is toprove4.5.(2)\Rightarrow 4.5.(3).(\mathrm{i}\mathrm{i})
.Again,
letf
bea Urieskornpolynomial
and definea_{n}by
\tilde{Z}_{f}1T
)=$\Sigma$_{n\geq 1}a_{n}T^{n}
. Assume that n\geq 1 isamultiple
ofaneven
exponent
off
butis notamultiple
ofanoddexponent. Then,
by
acloser lookattheconvolution
\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}_{J}
weget
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 ofaBrieskorn
polynomial
with evenexponents
from the virtual Poincarepolynomials
ofitspreimages
over1 and -1,wecould concludeby
induction. We believe thatit ispossible
sothat the
conjecture
isreducedto thecomputation
ofsomevirtual Poincarépolynomi‐
als. Noticeit
already
holds forahomogeneous
Brieskornpolynomial
ofevendegree
by
[10,
Corollary
2.5 &Corollary
2.6].
4.2
Arc‐analytic
classification of someweighted
homogeneous
polyno‐
mials
In the
complex
case, it isknown that the localanalytic
type
ofasingular
weighted
homoge
neous
polynomial
with isolatedsingularity
attheorigin
determinesitsweights
[24].
Italsoholds
by
considering merely
thetopological
type
in two[27]
and three[23]
variables.T.Fukui
[12,
Conjecture
9.2] conjectures
the realcounterpart
forweighted homogeneous
realpolynomials
with isolatedsingularity
intheblow‐analytic
context.Thisconjecture
has been provenby
O. M. Abderrahmane[1]
in twovariables andby
G.Fichou andT.Fukui[11]
inthree variablesinthe blow‐Nashcontextforconvenient
weighted homogeneous polyno‐
mials whichare
non‐degenerate
withrespect
totheir Newtonpolyhedra.
Since Brieskorn
polynomials
areconvenientweighted
homogeneous polynomials
whichare
non‐degenerate
withrespect
totheir Newtonpolyhedra,
it isnaturaltoask whether the materialpresented
intheprevious
sectionwould allowonetogeneralize
the result of G. Fichou andT.Fukui withnoconditiononthe number of variables.Afirst obstacleisthatwecantusethe convolution formula anymoresincewecantas‐
sumethat sucha
polynomial
isasumof pure monomials. However, it isstillpossible
toadapt
thestrategy
usedtoprove that thearc‐analytic
type
ofaBrieskornpolynomial
de‐termines its
exponents.
Itrelieson aformulatocompute
the modifiedzetafunction ofaJean‐Baptiste Campesato
Theorem
4.6([5]).
Letf,g\in \mathbb{R}[x_{1},\ldots,xd]
betwoarc‐analytically
equivalent weighted homogeneous
polynomials
whicharenon‐degenerate
withrespect
totheir Newtonpolyhedra.
Then1. Either
they
arebothnon‐singular,
andinthiscasethey
arebotharc‐analytically
equivalent
to(xl,\cdots x )
2. Or
they
share thesameweights(up
topermutation
andpositive
commonmultiplicativefactor).
Acknowledgements.
The authorissupported
by
aJapan Society
for the Promotion of Sci‐ence
(JSPS)
PostdoctoralFellowship
(Short‐term)
for North American andEuropean
Re‐searchers.
References
[1] \dot{\mathrm{O}}.M.AUDERRAHMANE,Weighted homogeneous polynomialsandblow‐analyticequivalence,inSingularity theory
anditsapplications,vol.43of Adv. Stud. PureMath.,Math. Soc.Japan, Tokyo,2006,pp. 333‐345.
[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 létude dessingularitisrédles,PhDthesis,Umversité Nice
Sophia Antipolis,122015.
[4] —,An inversemamping theoremforblow‐Nash mapsonsingularspaces,NãgoyaMath.J.,223(2016),pp.162‐
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.PARUSiNSKJ,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 darcs réelsetséne dePoincarédun 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
[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.