C OMPOSITIO M ATHEMATICA
F ÉLIX D ELGADO DE LA M ATA
A factorization theorem for the polar of a curve with two branches
Compositio Mathematica, tome 92, n
o3 (1994), p. 327-375
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A factorization theorem for the polar of
a curvewith
twobranches
FÉLIX
DELGADO DE LA MATAUniversidad Valladolid, Depto. de Algebra, Fac. Ciencias, 47005 Valladolid, Spain Received 12 July 1990; accepted in final form 14 June 1993
Introduction
Let
f E C[[X, Y]]
be a reduced formal series over thecomplex
field C(i.e.
f = II i f
where thefis
are irreducible andfi *fj
ifi -# j),
and leth E
C[[X, Y]]
be aregular parameter (i.e.
h defines anonsingular plane algebroid curve).
Thepolar
off
withrespect to h, P( f, h),
is thealgebroid
curve defined
by:
Examples by
Pham show that thetopological type
ofP( f , h) depends
on theanalytic type
ofC,
the curve definedby f
=0,
and notonly
on itstopological type,
even for h transversal tof
However we may wonder what information ofP( f, h) depends
on thetopological type
of C.Roughly speaking:
Assuming
thetopological
typeof
Cfixed,
What can we say about the
topological
typeof P(f, h)
The best results about this
question
have been obtainedby Lê,
Michel andWeber
([LMW], [LMW2])
for the case in which h is transversalto f, by using topological
methods: let n :X H e2
be the canonical resolution of the germC,
E theexceptional
divisor of n.Then, P,
the strict transform ofP( f, h),
does notmeet the strict transform of
C,
and one can determine(not completely)
thecomponents
of E which meetP.
As a consequence,they
alsocompute
the set ofpolar quotients,
thatis,
the setThis paper was partially supported by the CICYT: PB88-0344-C03-01.
where
( f, cp)
denotes the intersectionmultiplicity
off
with 9 andm(g)
isthe
multiplicity
of 9. Polarquotients
were introducedby
Teissier anddepend
on thetopological type
of Conly ([T]).
Similarcomputations
havebeen made
by
Kuo and Lu in[KL]
andby
Steenbrink and Zucker in[SZ].
When
f
isirreducible,
the first methods used tostudy
theproblem
abovewere of arithmetical nature. Merle in
[M]
proves that if h is transversal tof,
thenJ(f, h)
can be factorized asr 1 r 9’ where g
is the number of Puiseuxpairs
off
and eachri
is aproduct
of branches with constantpolar quotient.
Thepolar quotient corresponding
to branches ofri
and themultiplicity
ofri
can beexplicitly computed
in terms of the minimal set ofgenerators
of thesemigroup
of values off (the
factorization Theorem is alsoproved using
the arithmeticalproperties
of thissemigroup).
The i-thpolar quotient, corresponding
to a branch inri,
isequal
to thecoefficients of contact, ( f, Ç)/m(Ç),
for a curve03A8
of genus i -1, having
maximal contactwith
f,
thatis, tf¡
has the maximalpossible
intersectionmultiplicity
withf
among the curves with i - 1 Puiseux
pairs.
Fromthis,
one finds the set offree
infinitely
nearsingular points
that a branch ofFi
and the curvef
havein common
and,
as consequence, one obtains the results in[LMW2]
in amore
precise
form. Notice that the first i - 1 Puiseuxpairs
of a branch ofFi
areequal
to thecorresponding
ones forf Ephraim
in[E]
extendsMerle’s result to any
regular h, by using
the( f, h)-sequence
instead of theminimal set of
generators. Finally,
both results aregeneralized by Granja ([G])
to alarger
class of curves which includes thepolars
as aparticular
case. We want to remark that the
only
data used in the results above arethe intersection
multiplicities
ofJ( f, h)
withf
andh,
and no otherproperties
ofpolars.
The arithmetical
description
of thesemigroup
of values of a curve with several branchesgiven
in[D], (and
in[B]
for the case d =2)
induces usto
try
to obtain similar results for thepolar
of a non irreduciblef using
the arithmetic of the
semigroup
andtaking
asonly
data the intersectionmultiplicities
ofJ( f, h)
with the branchesfl, ... , fd
off
andh,
that isIn this paper we carry out this program for the case of two branches
(d
=2)
and h not
necessarily
transversal tof Unfortunately
this is notpossible
formore than two branches
(d > 3)
as theexample (4.12)
shows.So,
ford >
3 furtherproperties
ofJ( f, h)
are necessary in order togive
a"good"
factorization or in order to refine the results in
[LMW2].
The main results
proved
here go inparallel
with the onesproved by
Merle and
Ephraim.
There is also aninterpretation
in terms of theresolution process for the
singularity of f = 0,
included in section 4.Taking
into account that the
topological type
of a curvesingularity
is anequivalent
data to the
semigroup
of values(or
to the resolutionprocess),
thetopological type
ofP( f, h)
has some restrictionsimposed by
that one ofC,
as well as some "similarities" with it.
More
precisely,
assume for the sake ofsimplicity
that h is transversal to the curve with two branches C definedby f = f 1 f2
= 0(in
the paper there is no restriction forh).
Denoteby S - Z2
thesemigroup
of valuesof f
andfor any qJ E
C[[X, Y]]
letI(f, q)
=(( fI’ qJ )/m( qJ), ( f2, w)/m(w)).
If q is anirreducible
component
of J =J( f, h)
we will say that1 ( f; ç)
is apolar multi-quotient of f
Then J has a factorizationwith the
following properties:
The numbers s and t are the number of Puiseux
pairs
of the branchesdefined
by fl
andf2 respectively.
Each factor is aproduct
of branches of J with constantpolar multi-quotient.
Thepolar multi-quotient correspond- ing
to one of thefactors, r,
itsmultiplicity
and the value in S of I-’ thatis,
the orderedpair
of natural numbers((r, f 1 ), (r, f2))
- can beexplicitly computed
in terms of certain set of elementsof S,
called the set of values of the maximal contact. This setplays
a role similar to that of the minimal set ofgenerators
in the case of an irreduciblef (notice
that thesemigroup
of values of a curve with several branches is not
finitely generated,
but canbe "determined"
by
a finite number ofelements,
see[D]).
Given
9 c- F
ia q
+1,
the first coordinate ofI(f, 9), (fl,cP)/m(ep)
is thei-th
polar quotient
of thebranch f1 (the
second one can beeasily
determined in terms of the first one and
( fl, f2)),
so we can determine in aprecise
form the set of freeinfinitely
nearsingular points
in common for 9 andfl.
Similar results are true for the irreduciblecomponents
ofh2,
i
a q
+ 1. The irreduciblecomponents,
ep, of thefirst q factors, Fi (1
i xq), satisfy
and this rational number is
equal
to the i-thpolar quotient
off, (or f2)
sowe can make a similar
interpretation
in terms of theinfinitely
nearsingular points. Finally,
Dcorresponds,
ingeneral,
to the irreducible factors of J which have in common withf
the set of commoninfinitely
nearsingular points of fi
andf2.
In otherwords,
the branches ç of D become transversal tof’ exactly
in the samestep
of the resolutionprocedure
in whichf,
andf2
become transversal. There are some cases in which the existence of sucha D cannot be
guaranteed
because itscomponents
are, from thepoint
ofview of
arithmetic, undistinguishable
from the components ofrq
orFq’+ i,*
see Theorems
(3.11), (3.12)
and(4.9)
for aprecise
statementtaking
intoaccount all the
possible
cases.Now,
let n : X -->e2
be the minimal embedded resolution for the germC,
C’ the total transform of C and E theexceptional
divisor. We can translate the results above in these terms,obtaining
thecomponents
of Eappearing
in the resolution
procedure
for P =P( f, h) (not completely,
becauseonly
the free
infinitely
nearsingular points
aredetermined)
or, in otherwords,
the irreducible
components Ei
of E such thatEi
n P =1=0, P being
the stricttransform of P
by
n.Moreover,
as a consequence, thepolar multi-quotients
are in 1-1
correspondence
with the"rupture
divisors" of E(that
meansirreducible
components
F of E such that #F n(C’ - F) >, 3).
Infact,
theset of
polar multi-quotients
can be realized as theset {I(f, n(çF))}’
where Fbelongs
to the set of"rupture
divisors"and ÇF
is the germ of a smoothcurve in X with normal
crossings
with E at apoint
of F.Notice that the
point
of view of this paper is rather different to the oneof Casas in
[Ca].
He proves that iff
is a"general"
element in the set ofcurves with a
prefixed topological type,
then theequisingularity type
of thegeneric polar
can becompletely
determined.Briefly,
the contents of the different sections are as follows:In section 1 we fix the notation and prove some results
involving
thecoefficient of contact
(in
the sense ofHironaka)
of two branchesf and g
with
respect
toh,
thatis,
the rational number( f, g)l(g, h).
The mainarithmetical
properties
needed in thefollowing
sections are stated.In section 2 we prove a factorization theorem for curves g such that
( f; g) (for f irreducible) belongs
to anenlarged Apery
basis off
withrespect
to h(see (2.2)
for the definition of suchenlarged Apery basis).
These resultsare needed in order to prove a factorization theorem
(at
thebeginning
ofthe third
section)
for thepolar
of a curve with d branches in a veryspecial
case named here
diagonal
case(d
branches with the sameequisingularity
type
andhigh
intersectionmultiplicity
betweenpairs
ofbranches),
and alsoprovides
animportant
tool for theproof
of the factorization in the two branch case.Section 3 is devoted to
proving
the mainresults, namely
the factorization ofJ( f, h),
for a curvef
with twobranches,
intopackages
with constantpolar multi-quotient.
Finally,
the lastpart
is devoted to thecomparison
with the results in[LMW2]
when h is transversal tof
and to extend thesegeometrical
interpretations
toarbitrary
h. At the end of thispart
we include acounterexample
of the factorization theorems when one takes d = 3 instead of d = 2.1.
Preliminary
factsThroughout
all thispaper k
will denote analgebraically
closed field andh E k[[X, Y]]
an irreducibleregular parameter,
thatis, h
defines a nonsin-gular plane algebroid
curve over theground
field k.(1.1)
Let C be an irreduciblealgebroid plane
curve definedby
an irreduc-ible formal series
f E k[[X, Y]].
Denoteby S(C), S( f ),
orsimply
S ifconfusion is not
possible,
thesemigroup
of values ofC,
thatis,
the set( f, g) denoting
the intersectionmultiplicity
betweenf
and g, and v the normalized valuationcorresponding
toC,
thatis,
the valuation associatedto the valuation
ring t9,
normalization of U in its field of fractions.(1.2)
The maximal contact values ofC, flo, ... , 1 fi,, (see [Z], [C])
can bedefined as the minimal set of
generators
of S in thefollowing
way:and for i > 1,
It is well known
(see [Z], [C]) that, if {f3o,.", f3g,}
denotes the set of characteristicexponents
ofC,
then g =g’,
ei =
gcd(f3 0’
... ,f3J = gcd(fJo, ... , Bi),
These
equalities provide
theequivalence
between the two different sets ofexponents.
(1.2.1)
REMARK. Recall that in characteristic 0 the set of characteristicexponents
can beeasily
defined in terms of a Puiseuxexpansion
forf :
Assume that x is transversal to
f,
denoteby n
themultiplicity
off
and lety =
li> 0 aix’ln a
Puiseuxexpansion
forf
Then{30
= nand, recursively, {3i
is the minimum
integer
k such thatak :0
0 andgcd({3o, ... , {3i - l’ k) gcd(p 0,
... ,Pi ).
Theinteger g
isusually
called the"genus"
of the curvesingularity given by f
and3i
is thehighest possible
contact of
f
with an irreducible curve of genus i - 1. Inparticular el is
thehighest possible
contact off
with a smooth curve,fil = max{Ct:g) IgEk[[X, YJJ regular}.
When the characteristic of k ispositive,
sameproperties hold,
now in terms ofHamburger-Noether expansions
instead of Puiseuxexpansions (see Campillo [C]).
(1.3)
Let m be the intersectionmultiplicity
betweenf
and h: m =v(h) = ( f, h)
E S. Then(30 m B1,
and the m-sequence, vo, ..., v,, of S is definedas follows
(see [A], [E], [P]):
and for i >
1This
procedure stops
when we find sEN such thatds = 1. So,
vo, ... , vs}
c S anddo
>d,
>...> ds
= 1. Note that theflo-sequence (ob-
tained
taking h
transversal tof )
isexactly
the set of values of the maximalcontact. The m-sequence can be seen as a
system
ofgenerators
for S with respect to m(or h).
Associated with the m-sequence vo, ... , Vs we define the natural numbers(1.4) Depending
on the differentpossibilities
for m,namely,
m =kpo (1 , k [111/110])
or m =31,
therelationship
between the m-sequence and the minimal set ofgenerators
of S are as follows:(a)
If m = vo =30 (w
1 >vo),
then s = gand vi _ 31
for i =1, ... ,
g.(b)
If vo = m =kPo
for someinteger k
> 1(=>dl
1 -v 1 )
thens = g + 1,
v 1 =
flo and vi
=Pi - for
i =2, ... ,
g + 1. Notethat,
as consequence,The minimal set of
generators
of thesemigroup
of aplane
curve,( Bo, ... , P,,I,
satisfies some well-known andimportant properties.
The mainones for our purposes are
These
properties
characterize thesubsemigroups
of N which are thesemigroup
of values for someplane algebroid
curve. The first one is relatedto the
property
ofcomplete
intersection for a monomial curve, see Azevedo[Az], Herzog [He]
andAngermüller [An].
Similar
properties
can beeasily proved, using computations above,
form-sequences,
namely:
(1.5)
The contactpair, (f g),
for two irreducible branchesf
and g, is defined in[D, (3.3)]
in terms of theHamburger-Noether expansions
off
and g. This
pair
ofintegers, ( f g) = (q, c),
can be also characterized in thefollowing
way:Denote
by 11’0,..., B 9 eô, ... , e’, 9
the numbers defined in(1.2)
for thecurve g and let t be the minimum
integer
such that(Note
that thisinteger always exists, setting JJg+ 1 = JJgf +
1 = oo if it isnecessary.)
Denoteby lt (resp. lt)
theinteger part of (Pt + 1
-NtJJt)/et (resp.
of ( fl§ + i - N§ fl§)le§). Then,
- If
(1, g) p(t),
there exists aninteger
c, 0 cmin(i, 1§),
such thatIn this case q = t and
( f g) _ (t, c).
- If
( f, g)
=p(t)
ande; Pt + 1 #- et P; +
1 then(q, c) = (t, (0). (In [D]
thiscase appears as
(t, min {Zt
+1, lt
+1})).
Recallthat,
ifç
=e; Pt +
1et P; +
1then lt It
andif lt lt then ç
=e; Pt +
1et P; +
1([D]).
-
Finally,
if(1: g)
=p(t)
ande;pt+l
=etp;+I, (q, c) = (t
+1, 0).
Note
that,
if we set ,the
possible
intersectionmultiplicities
between two branches with the samesingularity types
asf and g (that is,
the same maximal contact values orthe same characteristic
exponents)
are the elements in the setwhere
ç(q, c)
=e’ - 1 pq
+ceq eq
=e. - , fl’
+ceq eq
andj(oJ)
=min{ e1fp+
1,ep 3p + 1 1.
These elementscorrespond
one to one to the set ofpossible
contact
pairs:
Moreover, (taking
thelexicographic
order in(N u { 00 } )2),
(1.5.1)
REMARK.Suppose k
of characteristic 0 and x transversal tof
andg, the contact
pair
can be characterized in terms of Puiseuxexpansions
forf and g
as follows: Denoteby n (resp. n’)
themultiplicity
off (resp.
ofg)
and let y =
li>-Oai xi In. y’
=Y- i >_ 0 a’x’ln ’ i be
Puiseuxexpansions
forf and g respectively.
Assume that
(f 1 g) = (q, c)
with c 00. Then there exists a n’-th root of theunit,
ev, such that ai =a’iwi’
forany
i such thati/n
y, wherey
(fi,
+ce,)In
=(f3q
+ce’)In’ and,
moreover, y is thegreatest exponent
with these conditions. If c = oo, the sameproperty
holds withy
= min{f3q+ lin, Bq+ i/n’);
in this casef3q+ lin =1= f3q+ 1 In’.
Theseproperties permit,
in the characteristic 0 case, to define the contactpair
and to prove in an easy way theproperties
above for it(see [Z2]).
As in the Remark(1.2.1)
thepositive
characteristic case can be stated in a similar wayusing Hamburger-Noether expansions ([D]).
(1.6)
Similar statements to those obtained in(1.5)
can begiven taking
them-sequences for
f
and g withrespect
to a fixedregular parameter
h instead of the sets of values of the maximal contact. Denoteby vo, ... , vt, dû, ... , dt,
... thecorresponding
datafor g
withrespect
to h and let r be the minimuminteger
such thatThen we can define the contact
pair
off and g
withrespect to h, (f g)h,
and also prove the
following:
(1.7)
LEMMA-DEFINITION. With notations as above:such that
In this case
define (f g)h = (r, d).
(3) If ( f, g)
=t(r)
anddr Vr+
1=1= dr Vr+vr
1 thenÔ’r + 2 #- br+
2, and wedefine (f 9)h = (r, c)o)-
(4) If (1, g)
=t(r)
andd’ rVr+ d rV’r+ 11
thenMr +
iMr + 1, Ô’r + 2
=br+2
and
dovr +
1= d’ovr+
1 1 n this case,define (f 19) h
=(r
+1, 0).
(5)
Therelationship
between(f [ g) = (q, c)
and(f g)h
is asfollows:
If q >, 1,
thenProof
Theproof
of(1)
is asimple computation taking
into account therelationship
between the m-sequence and the values of the maximal contactgiven
in(1.4)
and(1.5).
For the rest of statements, note first that in the case in which h is transversal tof and g
the result istrivial,
because(f g)h
=( f g).
So,
assume that h is not transversal and denote(f 1 g) = (q, c).
We shallconsider three different cases,
depending
on q and c.In this case, as h is
non-singular, (f h) = (g h) (f g). Thus,
there are twopossibilities.
The first one is( f, h)
=k11o, (g h)
=keo
with 1k lo
=l’o.
Inthis case one finds
(f g)h
=(q
+1, c)
and the result is a consequence of(1.5).
In the other case, one must have
(f, h) = fi,, (g, h) = fi’
and then( f 1 g)h
=(q, c),
so the result istrivial,
sinced 0 V1 _ 1 /3o
=e’o 111
=eo fi -j
=v 1 d’o.
Without loss of
generality
we can assume that( f, g)
=e’o1Jl’ so % l’o.
Thedifferent
possibilities
for h are thefollowing:
In this case,
( f, g)
=cio 3
anddepending
on h we have thefollowing
possibilities:
. This is the same case as above with
f
andg
interchanged.
(1.8)
REMARK. Let r be as in(1.6),
the minimuminteger
such that( f, g) t(r). By
the Lemma above we haveand,
r is the minimuminteger
such that theinequality
above holds.Moreover, dr
divides( f, g) except,
at most, in the case( f, g) = dr vr +
1dr vr + 1,
thatis, except
when:As
Consequence,
Finally, then,
sincemust be r > n and so
bn+l
=Ô’n+1-
(1.9) Let f,
g,p E k[[X, Y]]
beirreducibles, keep
the notations above for the different invariants associated tof and g
and denoteby Bo", .... eo, ... ,
thecorresponding
ones for the curve definedby
p. It is well-known that at leasttwo of the numbers
are
equal,
the third onebeing greater
orequal
than therepeated
one([Pl]).
In other
words,
This fact can be
easily generalized changing
themultiplicities
of the curves forthe intersection
multiplicities
with h. Moreprecisely:
(1.10)
PROPOSITION.Let f,
g,p E k[[X, Y]] be
irreducibles. AssumeThen,
Proof
Consider the setBy (1.9)
this set has at most twoelements,
so we have thefollowing possibilities:
By hypothesis (p, h)(f, g) > ( f, pX g, h)
and hence(p, h) > ( f, p)(g, h) ( f, g)-’ - ( f, p)e 0 1. Using (1.9)
for p, hand f,
we findso,
( f, p) = ( f, h)e’
andby (1.9) again (g, p)eô
=(g, h)e’e",
that isAs
above, by hypothesis ( p, h)
>( f, p)(g, h)(f, g) -1
1 >( f, p)e ô 1,
and thenAs consequence,
(g, h)e’Ó(e’o) -
1and by ( 1.9)
Assume
( f, g)e" ( f, p)e’o; using (1.9) for f, p
and h we obtainBut in this case,
which
gives
a contradiction. As consequence( , f; g)e’b
>( f, p)e’o
and( g p)ei = (g, p)eo.
Thisequality, together
with( f, h)e’o
=(g, h)e o
leads to the result.(1.10)
REMARK.Obviously,
theproposition
above isequivalent
to say that in the setthere are at most two different
elements, being
therepeated
element the minimum of the set.2.
Apery
basis and curves with contact in itLet
f E k [ [X, Y] ]
be an irreducibleseries, h E k [ [X, Y] ]
irreducible andnon-singular
and m =( f, h).
Wekeep
the notations of section 1.(2.1)
DEFINITION. TheApery
basis of S withrespect
to m is the orderedset
The
Apery
basis has been treatedby
several authors([Ap], [An], [A],
[P], ... )
and some of the facts that we will use below can be found in these references.Let
{vo,..., vs} be
the m-sequence ofS,
then one cancompute
theApery
basis in the
following
way: If k is aninteger
with0 k , m - 1, k
can be written in aunique
way as k= Li liibi
with a;integers
such that0 ai Mi (1
is). Then,
(2.2)
DEFINITION.Let ç ds-l Vs = 11
be a natural number. The en-larged Apery
basis of S withrespect
to mand ç
is the ordered setAm,ç = {aillE N}, where,
if 1 = pm+ k
with0
k m wedefine a,:= pç
+ ak.(2.3)
REMARK. We haveenlarged
theApery
basis in such a way that theequivalence (2.1.1 )
is conserved. Infact,
if l EN,
l can be written in aunique
way as
l = pm + LÎ lJ.i5i
with0
ociMi (1 -
is)
andIn the
following,
we shall prove some facts for theenlarged Apery
basiswhich are in
general,
known for theApery
basis. Unless otherwisespecified, ç
will be a natural numberwith ç r
=ds _ 1 vs,
and the elements a,,(1 EN),
will be the elements of the
enlarged Apery
basis of S withrespect
to m andç.
(2.4)
LEMMA. Letl1, ... , lt
be natural numbers and 1= Ltl li. Then, Ltl
ail ai.Moreover, if li
mfor
all i =1,...,
t and 1 = pm + k withk m then Y-’ 1 a,, , pq + a k
Proof
We will use induction on t. In the case t = 1 there isnothing
toprove.
So,
assume t >1, put
l’= Li-l li
= qm + v with v m and assumethat £b ai, qç
+ au(resp.
qn + au ifli
m for anyi).
If
lt
= rm + u with u m, thenThus,
theproblem
is reduced to thecomputation
of a,+ au when v,
u m.Assume u
== Y-’ uibi, v == Z’ Vi5i
with Ui’ ViMi, (1
is).
Then we canwrite:
where yi Mi
and ei c-{O, 11
for i =1,...,
s. We maycompute
u + v andau + av:
Note that k =
L Yibi,
as Y-yi ôi
m and l == Eyi ôi (mod m).
On the other hand:
This
computation provides:
(2.5)
REMARK. This lemma abovegeneralizes
the well-known fact that a"+ au C
au + v if u + v m(see
the references in(2.1)).
Note that we havealso
proved
that:. If ç > 1], av + au = au+v=>Bi = 0,
1 i s; inparticular
u + v m.If ç = 1], av + au = au + v => Bi = 0, 1 i s - 1.
These facts can be
easily generalized
to the case of anarbitrary number,
t, of summands:
Let
l 1, ... , lt
E N and 1= Ltlli’
Assumeli
=Y- 1 =: 1 Dc’ôj i
+ Pi M withoc i Mj
for 1
j,s and
1 i t and1=1’=,Lx,,ô,
with xrMr
for 1 r s.Then one has:
and in this case must
be, Xl) = i a§ = rij
for 1j s
andXl) =
1 pi = p.(B) If ç = 1],
and in this case must
be, Xl)=i a§ = rJ.j
for 1K j K s -
1 andXl) = i ( p; + OEL) # p + OE.
(2.6)
PROPOSITION.Let g
ek[[X, Y]] be a formal
series and suppose thatfor
each ç, irreducible componentof g,
we haveThen
(g, f ) - a(g,h)’
Proof. Let g
=II1
gi be the factorizationof g
in irreducible elements andset li
=(Pi’ h). By
Lemma(2.4)
it suffices to prove that(Pi’ h) - a,,
for anyi =
1, ... , t,
thatis,
we can assume g irreducible.Denote
by v’, ... , v’,, d’, ... , ds, ...
the data defined in(1.3) for g
withrespect
to h and let r be the minimuminteger
such thatalso define
If r = s,
then(g, h)
=d o
=d{f, h)
=d’ s m
=S m (in particular
k =0)
andby
thehypothesis
Otherwise,
r s and we haveas we want to prove.
(2.7)
REMARK. TheProposition
above is known for theApery
basis.Note
that,
if(ç, h)
m,automatically
From the
proof of Proposition (2.6)
and Remark(1.8)
we can deduce theconditions in which the
equality (g, f ) = a(g,h)
holds.Assume g
irreducible and write(g, h)
= pm + k = pm+ Li Yi ôi with oci Mi (1
is).
(A)
If(g, h) >
mand ç
> q, then:(B)
If(g, h) > m and ç =
q, then:Note that in this case
( f, g) = ds _ 1 vs
andd’ , - 1 = PMS + Ys ôs = pMs
+ k.(C)
If(g, h)
m, then:(g, f )
-a(g,h) =>
3r suchthat xi
= 0 for alli =1= rand (.f; g)
= xrvrAs consequence, among the numbers xl, ... , xs there exists at most one not
equal
to zero, say Yi., and the conditionYiop *
0implies that ç = 11
andi o
= s.(2.8)
THEOREM. Letf,
g Ek[[X, ¥JJ; f being irreducible,
and supposefor
each irreducible component çof
g. Let l =(g, h)
= pm + k = pm+ Li a;à;
with xiMi, If (g, f )
=a(g,h)
=pç
+ ak,then g
can befac-
torized as g =
Al A2 ... AS B in
such a way that:(3) If ç
> il then(B, h)
= pm and(B, f)
=pç.
(4) If ai -=1=
0(resp. if
B =1=1),
thenfor
any irreducible component,Aij (resp. Bj), of Ai (resp. of B)
we haveProof Let g
=II i
çi be the factorizationof g
in irreducible elements.Denote li
=(CPi’ h)
= Pim+ Li a§à with a § Mj
for i =1,..., t; j
=1,..., s.. By
the resultsabove,
we have(p i, , f’)
= ail for any i =1,..., t,
andby (2.7)
this occurs if andonly
if:(1) rtj
= 0 forall j except,
at most, for anindex j(1).
(2)
Ifp; #
0 thenj(i ) = s and, either ç
= 11or a§
= 0 for everyj.
(3) (cp i’ f) = Pi ç + a§;>vj;>.
.
If ç > 1]
defineA() ={!,..., t} - Uî A(k)
and ifç = N], A(s) =
( 1, ... , tl
-Uî-1 A(k).
Note thatNow,
we defineFor k s
(resp. k
sif ç
>il)
we haveand
by (2.5)
theinequality
in(*)
is anequality
if andonly
ifY-ic-A(k) y k iM k
.Now assume
that ç
= ~]. Thenand,
sinceLiEA(k) (i Mk,
we haveLiEA(k) C( = C(k
for any k s and(L pJm + £ a(
= pm +as às .
As consequence,(A k, h) = £; akà = Ckbk
andalso
(Ak, f )
=03A3i xk vk
= (ik Vk for k s. The last statement in the Theorem for thecomponents Aij of Ai
is the Remark(2.7) above, part (C)
if i sand
part (B)
if i = s.Finally, write £ a(
=p’m + xs with xs MS . Then Xp1
+p’ =
p,xs
= lis andas we wanted to prove.
The
case ç
> il can beproved
in the same wayusing
the Remark above and similarcomputations.
(2.9)
REMARK. Theorem(2.8)
is well-known in someparticular
cases:Merle in
[M]
proves this Theorem for the case in which m= 30, ( f, g)
= am _1 and (g, h)
= m - 1 with the maingoal
of the determination of thepolar quotients of f. Ephraim
in[E]
proves the same result as Merle for anarbitrary
m =( f, h). Finally,
the same Theorem has beenproved by Granja ([G])
for theApery
basis withrespect
to m.In the rest of this Section we shall
give
someapplications
of(2.8)
to asimple
case of curves with several branches.(2.10)
APPLICATION:Diagonal
curves with several branches. Letf = II 1 f E k[[X, Y]]
be suchthat fi
is irreducible for anyi EI:= {l,... , d}
and fi * fj
if i=1= j.
Denoteby
vi the normalized valuationcorresponding
tothe
branch f
and let «-) =k[[X, Y]]/( f )
the localring
of theplane
curvedefined
by f
Thesemigroup
of values off
is thesubsemigroup, S,
ofNd given by
S:=
{12(g):= (v 1 (g),
...,Vd(g» 1 g E (9,
g non-zerodivisorl.
We shall say that
f
isdiagonal
withrespect
to h if thefollowing
conditions are satisfied:
(1)
The m-sequence forf
withrespect
to h isindependent
of i(1 ,
1 Kd).
In that case, denote itby
m = vo, ... , vs . Inparticular
this fact
implies
thatthe fi ’s
areequisingular.
(2) (fi, fj) = ç
=ds _ 1 vS
+ cindependently
ofi, j E l.
(2.11)
From thedefinition,
iff
isdiagonal
then theelements Vi (v i’ ... , viJ E Nd (i
=0, ... , s), belongs
to thesemigroup
of values off
andthe same is true for 03C8 =
(ç,
... ,ç).
Note thatVo,... , V,,
03C8 areplaced
in thepositive part
of thediagonal
ofNd.
This set of elementspermits
thecomplete computation
of S(see [D], [Ga])
and these are the reasons forthe name
diagonal.
Consider
g E k [ [X, Y]]
such that:where p
d and aiMi
for i =1,...,
s.The
following
resultsgive
adecomposition
Theoremfor g
similar to thatone in
(2.8).
(2.12)
LEMMA. Letf
bediagonal
with respect toh,
and g as in(2.11.1). If
ç is an irreducible component
of
g thenand as consequence
(9, fi)
=(cp, fj)
Vi, j E
J.Proof.
First ofall,
if theinequality (*)
is true,then, taking
into accountthat ç = (i’ï,fj)
and( 1.10)
we find that(fj, 9)(fi, h) > (fi, 0)(fj, h),
orequivalently (fj, ç) a (fi, 9).
But the roleof f
andfj
can beinterchanged
and so
(fi, cp)
=(fj, cp)
for anyi, j E l.
We shall prove the first statement in the Lemma
by
induction on d. Thecase d = 1 is trivial
(see (2.7)),
so assume d > 1and, also,
that there exists cp such thatLet r be the minimum
integer
such thatvl«p) dr(p)v r+
1. Then if r s,Thus,
there must be r = s. As consequencebs+
1 =bs+ 1(q» and,
sincewe have
ds(cp)
p.For i =
2,..., d,
we have(cp, Il)(h, h)
>( f1, fi)(9, h)
and thenNow,
considerg’ = glg
andp’ =
p -d,(9).
Thenand we can use the induction
hypothesis,
so all the irreduciblecomponents
t/1
ofg’ satisfy
By (1.10)
there must beV1 (tf) v; (03A8)
and thenBut,
in this case,and we reach a contradiction.
(2.13)
THEOREM.Let f
bediagonal
with respect to h and let g be as in(2.11.1). Change
in Theorem(2.8)
viby V:, ç by
IF and( f - ) by _v( - ).
Thenthe
resulting decomposition Theorem for
g is true.Proof.
It is a consequence of(2.12)
and(2.8).
3. Factorization of the
polar
of a curve with two branchesLet
f = n1 f E k[[X, Y]]
be such thatf
is irreducible for any i EI:={1, ... , d 1 and fi * fj
ifi * j,
and leth E k[[X, Y]] defining
anon-singular algebroid
curve. Denoteby Vi
the normalized valuationcorresponding
tothe
branch f
andby So... v’, 0
... ,vsl, diô, ... , disi, ...
the data definedin
(1.2)
and(1.3) for f
withrespect
to h( 1 i d). S
stands for thesemigroup
of values off and Si
for the one off .
(3.1)
POLAR CONTACTS. Assume for a moment that k is of character- istic zero. Thepolar
curve off
withrespect
to h is thealgebroid
curvedefined
by:
and it is well-known
(see [M], [E])
thatwhere ci is the conductor of the
semigroup
ofvalues Si
off . Moreover,
and then
By [D] (2.6)
and(2.7)
the conductor b of thesemigroup S (that is,
the minimum element b E S such thatb + Nd S)
isgiven by pri(b) =
ci +y_j # i (fi, fj)
and the element i = ô -(1,..., 1) belongs
to S.Using
this notationwe find that
A
straightforward computation
shows that(3.2)
DIAGONAL CASE. Now assumef diagonal
withrespect
toh;
then theelements Vi = (v!,..., vi) = (vi,..., vi) belong
to S and the formulae aboveprovide:
As consequence, Theorem
(2.13) gives
adecomposition
for thepolar
curveJ( f h)
in thespecial
case in whichf
isdiagonal
withrespect
to h.(3.2.1)
REMARK. In thegeneral
case(that is, f
notnecessarily diagonal),
wecannot use
directly
the results of Section 2 in order togive
adecomposition
ofJ( f, h).
In fact the elements(vl1 vf)
do notbelong,
ingeneral,
to S and alsoMik Mli if k # 1. So,
the firststep
will be togive
an arithmeticaldecomposition
of T +
Va
in"good"
elements of S and then to use the arithmeticalproperties
of the