2. PUISEU X EXPANSIO N 3
which contradict s th e hypothesi s tha t / * i s irreducible .
/ ' therefor e define s a branc h C" , calle d th e quadratic transform o f C. On e ha s
e(C") e(C). Th e Theore m o f Resolutio n o f Singularitie s o f plan e curve s affirm s
that afte r a compositio n o f finitely man y quadrati c transformation s on e obtain s a
branch C wit h e(C ) = 1, that is , C i s nonsingular .
Thus, t o a give n branc h C i s associate d a sequenc e o f quadrati c transform s
C, C",..., C M , . . ., suc h tha t e(C ) e(C" ) .
We set e(CW ) = e$ . To any branch, on e has therefore attache d a monotonicall y
decreasing sequence of integers that equal s 1 beyond a certain index . Le t N denot e
the smalles t inde x suc h tha t e ^ = 1 and e/v- i ^ 1. Let e*(C ) denot e th e first N
terms o f th e sequence . W e ca n the n define , formally , th e equisingularit y o f tw o
algebroid branche s C an d D.
DEFINITION
1.5. C = D i f and onl y i f e*(C) = e*(D).
In th e comple x case , the tw o definition s o f equisingularity coincide , an d thi s i s
a nontrivia l result .
As in the convergent case , one introduces i n an equivalence class L o f algebroi d
branches the relation of analytic isomorphism (formal) , an d one obtains the modul i
space Lj = .
Our remar k i s th e following . Th e passag e t o th e forma l domai n i s onl y a n
apparent generalizatio n o f the concept . Thi s i s due t o th e fac t that :
(a) ever y algebroi d branc h (resp . analytic ) i s isomorphi c formall y (resp . an -
alytically) t o a n algebrai c branc h (tha t is , define d b y a polynomial ) (se e
[S]);
(b) i f tw o analyti c branche s ar e formall y isomorphic , the n the y ar e als o ana -
lytically isomorphic .
As a result, th e forma l situatio n i s not mor e general, an d i t suffice s t o conside r
the analyti c (o r even algebraic ) case .
2. Puiseu x Expansio n
We no w addres s th e followin g problem : construc t a mappin g fro m L(C) int o
a finite dimensiona l affin e spac e i n suc h a way tha t tw o branche s havin g th e sam e
image ar e analyticall y isomorphic .
In th e imag e o f L(C) i t wil l be necessar y t o identif y certai n point s i n orde r t o
give a description o f the modul i space ; nonetheless, b y this constructio n w e obtai n
a metho d t o provid e ou r modul i spac e wit h a topological structure .
Recall of some algebraic facts: Le t C b e a branc h wit h equatio n /(X , Y) = 0
where coordinate s hav e bee n chose n s o tha t f
n
(X,Y) Yn. On e ca n the n appl y
the Weierstras s Preparatio n Theorem . U p to a unit, / i s therefore a polynomia l
/ = Y n + A
1
{X)Yn~1+ + A
n
(X),
where Ai{X) £ C[[X]] . Moreover, eac h Ai(X) ha s orde r strictl y large r tha n i sinc e
C ha s multiplicity n an d f
n
= Y
n.
A s recalled i n §1, the Weierstras s polynomia l i s
also irreducibl e a s a polynomial wit h coefficient s i n K = C((X)) .
Let O = C[[X , Y]]/(f) = C[[x, y}] wher e x, y ar e the classes of X, Y i n O. Sinc e
/ i s irreducible , i t follow s tha t O i s a loca l rin g an d a n integra l domain . I n th e
following, O i s called th e local ring of the branch C.
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