Lecture
In this section we shall become acquainted with one of the most fundamental results of singularity theory, the Malgrange preparation theorem? (hereafter MPT), and obtain with its help some
consequences that will be used in what follows.
Let } (т): В” > В" be a smooth mapping, carrying the origin of
coordinates 0 into itself, which is given by п smooth functions:
(в) = (11(2),....Г“2)), т=(а,..02), №0) =0. (1)
1 Теап-С1аш4де Тоцрегоп — a French mathematician.
?Вегпаг4 Машгапяе (b. 1928) — a French mathematician.
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We shall denote by А the algebra of formal power series or
the algebra of germs of smooth mappings at the point 0. Accordingly, below we shall give the statement of the theorem in two categories: formal
series and smooth germs.
THEOREM 7.2 (MPT). Suppose that the mapping {(х):
В" — Е" has at 0 a critical point of multiplicity 1 < и < ©.
Let е\(х),...,е,(2} be generators of its local algebra ©;. Then
any element | (т) Е А can be represented in the form
(2) = а1({(2))е1 (2) --- + аь(Д(т))е» (5) (7.8)
with some suitable elements а; (т) Е А.
The proof in the category of formal series is not difficult, see .
In the category of smooth germs, the proof is based on a deep algebraic result about finitely generated modules (Nakayama's lemma), but also requires certain special results about smooth functions, see [9, 15, 30].
EXAMPLE 7.1. As a simple exercise let us see what
MPT gives in the case when и = 1, i.e. when the germ of the mapping } is a diffeomorphism. When и = 1 the functions {1(т),... {"(т) have at
the point 0 linearly independent gradients. Using a generalization of the Hadamard lemma (problem 1.1), we represent an arbitrary germ 9 Е А in the
form
9= 9(0) + В1 (=) {1 (2) +... + 8, (т) 1" (2)
with suitable 5;(т) Е А. Comparing this representation with (7.4), we
see that the function е1(т) = 1 is a generator of the local algebra О; (we could have taken as generator е!(15) any
constant function other than zero). Consequently, in our case
representation (7.8) takes the form
^(=) = а1(/(2))е1 (2) = а1(1(х))
and carries no information: since И is a local diffeomorphism, for any function Й Е А the equality № (т) = а1(Л(х))
holds with the function а1 (у) = (7 Т(у)).
The trivial example considered above should not discourage us: after all we have not yet applied MPT to critical points of
mappings. To this we now turn.
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PROBLEM 7.12. Prove that the germ of any smooth function
й(т,у1,...,Уп_1): В" > В can be represented in the form
р(х, у) — @1 (2, у) + га2 (27,5), (7.9)
where у = (у1,...,Уп—1) and а1.2(-, -) are certain smooth functions.
SOLUTION. In problem 7.4 we considered the germ of mapping (7.5). It was shown that the local algebra (©) of such a germ
is two-dimensional and has generators е1(х,у) = 1, е2(5, у) = х. From this, according to MPT, for any smooth germ А(т,у) we obtain the representation
№(х, у) = а1(1(т,у)) + хаз(Т(т,у))
with suitable smooth germs а1 2(., .), i.e. formula (7.9). ||
PROBLEM 7.13. Prove the following generalization of formula (7.9):
for any integer р > 2 the germ of any smooth function
(т, у1,..- вт): В" ЕВ
can be represented in the form
(2, у) = а1 (27, у) + таз(2?, у) +... + 7 Пар(т?, у), (7.10)
where у = (91,...,Уп-1) and а;(., *) are suitable smooth functions.
PROBLEM 7.14. Apply MPT to the germ /(х,у) : Е? -+ 82, given by the formula:
Л (х, у) = +(узу+ту Р(еу)=у, (7.11)
where 12(у) is an arbitrary smooth function.
SOLUTION. Let us show that the multiplicity ’/ = 3 and the local algebra ();
of germ (7.11) is generated by the generators
е1(х,у) =1, е2(х,у) =, ез(т,у) = Са. (7.12)
Indeed, using the Hadamard lemma and representation (1.5), one can represent any germ 9(т, у) Е А in the form
9(х, у) =о@1 + а 2х - азт? Е УВ (т, у) =Е хз Во(т, у),
where 1 = 9(0), а2 = 9(0), аз = 1 та (0) and В1.2(х, у) Е А, i.e. in the form
9(т, у) —
= а + а2х + аз? + уВ (т, у) + (23 + %(у)т?у + ту) Во(т, у),
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where В; = В: — (1 + 24(у))В2.
This is exactly representation (7.4) with the number и = 3 and generators (7.12). Hence, according to MPT, for any germ #(х,у) we obtain the
representation
(т, у) т а1({', у) -о хаз(} } У) + 2?аз(}', 9), (7.13)
where the function {1 = 1 (т, у) is taken from (7.11). [|
PROBLEM 7.15. Using similar reasoning, establish
representation (7.13) for the germ of the function {(х, у): Е? -+ В2, defined by the formula
(т, у) = ф(т,у) (23 +ч(уту+ ту), Г(т,у)=у, (7.14)
where ф(т, у) is a smooth function, $(0, 0) = 0.
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