Lecture
Let }(х): В" > Е" be a smooth mapping; it is given by a collection of т smooth functions: }'(т): В" -» В. Recall that the Jacobian
matrix of the mapping { is called the matrix „+, made up of the partial derivatives of the functions }\,..., {" with respect to the variables 21,..., 2”:
1 1. |
21 то и 2 2... 2
21 го С Л; = (7.1)
ить т... тт
21 шо РС
In the case of a mapping between spaces of the same dimension the matrix (7.1) is square, and its determinant is called the Jacobian of the mapping {.
DEFINITION 7.1. A point то of the preimage space is called a critical (variants: singular) point of the mapping
Г, if the rank of the matrix „/+ at this point is less than the maximum possible value ти {т, т}.
In the case п = т this is equivalent to the vanishing of the Jacobian of the mapping at the given point. If т = 1, then we get the already familiar
notion of a critical point of a function. If п = 1, then the mapping | defines a curve in the space В”, and this is the definition of a
critical point of a curve, introduced by us in section 6.
Recall also that in section 2 we introduced the notion of the multiplicity
of a function of one variable. Now it is time to define the general notion of the multiplicity of smooth functions ЕВ" -» Е and of mappings
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№" -› В". All subsequent definitions and results are of an exclusively local nature, and we shall speak of germs of functions and mappings at a chosen fixed point of the preimage space, which for simplicity we take to be the origin 0.
7.1. Multiplicity of mappings В" — В"
Let /(т): ЕВ" > В" be a smooth mapping taking the origin
of coordinates 0 to itself. It is given by п smooth functions:
(г) = (1 (+),...,/(2)), «= (ат, ,), Р(0)=0. (7.2)
Let А be the algebra of formal power series:
А=ЕВ[[1,...,5.]|.
Consider the ideal [; С А, generated by the functions /1(т),..., {" (т),
or more precisely, by their Taylor series at the point 0.
DEFINITION 7.2. The ideal [, is called the local ideal
of the germ of the mapping /. The local algebra of the germ of the mapping { is called the quotient algebra
С; '— А/Т,,
and its dimension х = Чпа @; is called the multiplicity of the germ }.
REMARK 7.1. The local algebra and the multiplicity are defined analogously
for germs of analytic mappings } : С” + С”,
one need only take the algebra А = С]т1,...,2и|] and carry out an analogous construction over the field of complex numbers. In the complex-analytic category there are many remarkable properties for which there are no analogues in the real category, see . For example, finite-dimensionality of the local algebra О; in the complex case is equivalent to
the condition that the point 0 be isolated in the set }—1(0).
Let us return to the case under consideration, of germs of smooth mappings В" -› В". A substantial theory exists precisely for germs of finite multiplicity. In what follows, unless stated otherwise, we
shall assume that и < со.
DEFINITION 7.3. A set of generators of the local algebra ©; is a collection of elements
е1(т), ..., е,(т) Е А, (7.3)
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which, upon factorization by the ideal [;, becomes a basis of the vector
space С).
Definition 7.3 means that for a fixed collection of elements (7.3) of the algebra А, every element 9(х) Е А can be represented in the form
9(т) = але (2) + --- + аье, (т) + В1(х) Г (т) + --:+ В, (2) 1" (=) (74)
with suitable numbers ©; Е ® and elements 0;(15) Е А.
Now let us take as А the algebra of germs of smooth functions
of п variables at the point 0. Consider the ideal /; С А, generated by
the functions ]"(т),..., 1" (т) and repeat all the previous reasoning
that we used to construct the local algebra С);. As a result we obtain the notions of the local algebra ©; = А/1; and the multiplicity
и = 41 О; in two categories: formal series and smooth
functions. There holds a remarkable fact about the coincidence of multiplicities
in the category of formal series and the category of smooth functions:
PROBLEM 7.1 (*). Show that if the multiplicity и, computed in
one of the two categories, is finite, then the multiplicity computed in the other category coincides with it. Moreover, the local algebras in both
categories are isomorphic: the generators of (С); in the category of formal series are obtained from the generators of ©; in the category of smooth functions by
assigning to each function }' its Taylor series.
In what follows we shall deal with germs of finite multiplicity, so there is no need to specify in which
of the two categories the local algebra and the multiplicity are defined.
PROBLEM 7.2. Let us make a smooth change of coordinates (т1,....Ть),
leaving the point 0 fixed. How will the local
algebra ©; and the multiplicity и change?
PROBLEM Т.3. Verify that if О is a regular point of the mapping ](1), i.e., the Jacobian „/+(0) 52 0, then its multiplicity и = 1.
Solution. By what was said above, it suffices to establish
the representation (7.4) with и = 1 either in the category of smooth germs or
in the category of formal series. Let us establish it in the category of smooth
germs, since for formal series the argument is even simpler. From the
condition „Л,(0) =^ 0 it follows that the functions }1(т),..., " (2) have linearly independent gradients at 0. Using a generalization of Hadamard's lemma
(problem 1.1), we represent an arbitrary germ 9 Е А in the form
9= 9(0) + В1(з)/* (=) + ---+ В» (=) (+)
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with certain smooth functions 2; (т). Hence, the local
algebra @; has one generator (for instance, one can take
е1 (2) = 1 as such), so its dimension и = 1. [|
PROBLEM Т.4. Compute the multiplicity of the mapping
В Иен, ЕВ >В"
at the point 0, given by the formula
(ту) и, ..., 1-Цх, у) — Уп-1, Е’ (т, у) = =”, (7.5)
where у = (91,..., м1), and find the generators of its local algebra.
SOLUTION. Let us show that any element 9(5) Е А can be represented in
the form (7.4):
9(5,у) = 90 (у) + 29% (9) + 27 92($, у) =
п-1 в-—1
= 90(0) + У. доу): + т (© В ри ии + 2*92(2,у) =
1=1 1=1
в—1
= 90 (0) + 291 (0) + У Ви(х, ууу: + Вы (х, у)? =
1=1
=а1 + а25 + У `` ВЕ(х, у)1"(х, у),
+=1
where @1 = 90 (0) — 9(0, у), @2 — 91(0) = 9=(0) and
В: (т, у) = до (у) + т9ы(у), +=1,....в-1; Вит, у) = 92(%,у).
In deriving this representation in the category of smooth germs
we used Hadamard's lemma and the representation (1.5) ср = 2 that follows from it. In the category of formal series, however, this representation is entirely obvious, since formal series can be
handled just like polynomials. In both categories we
obtained the representation (7.4) with multiplicity и = 2 and generators
е1(х, у) =Ь е2(х, у) — т.
Hence, the local algebra С); of the germ of the mapping (7.5) at 0
is isomorphic to the algebra of truncated polynomials В |/(22). №
PROBLEM 7.5. Compute the multiplicity of the mappings ®? -+ В? at the critical point 0, given by the following functions:
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$ (ту Р (т, у) 11| а+у2 у
7. ту у
3 ту т + у
4 с г
5 1:3 у
[ Ш 1 созу — 1
7| зщ (25) 11 (соз (7)
8 | зш(т фу) | № (со (2 -у))
7.2. Multiplicity of functions К” —} К
Let }(х) : Е" + В be a smooth function of п variables and 0
its critical point, i.e. },, (0) = 0 for all indices 1 = 1,...,п.
The function { generates the gradient mapping \/} : В" - В", defined by the formula
У/: т= (11,...,2в) = (Да. (т),..., (т). (7.6)
As for any smooth mapping В” -»› В”, for the gradient mapping \}{ one defines the notions of the local algebra
Оу; = А/Гу; and the multiplicity и = а Фу:.
DEFINITION 7.4. The multiplicity и of the function } : ®" -} В at the point 0 is called the multiplicity of the corresponding gradient mapping У} : В" — В", i.e. д = аа Оу.
REMARK 7.2. In the case п = 1 we have several definitions of the multiplicity of the germ {(т) : В! -+ В!. First, this is the multiplicity
и = 41 @; a particular case of definition 7.2. Second, according to
definition 7.4, this is the multiplicity д = апп @у;. Third, for functions
of one variable the multiplicity + was defined earlier by condition (2.1).
PROBLEM 7.6. Verify that for п = 1 the multiplicity д in the sense
of both definitions coincides, and moreover и = и + 1.
Solution. The equality и = д-+1 is obvious. Let us show that if the number
и is defined by formula (2.1), then the multiplicity of the function in the sense of definition 7.4 is also equal to д (the converse statement is proved analogously). From (2.1) it follows that the Taylor series of the gradient mapping
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УГ: тн+ ] (1) begins with the monomial т*. Hence, any element
9 Е А can be represented in the form
9(=) = Р,-1(=) + В(т)7 (=),
where Р,_1 is a polynomial of degree д —1and ВЕЛ. (For А =Е[ | this representation is completely obvious, while for the algebra of smooth germs one must use Hadamard's lemma.) It follows that the quotient algebra А/Гу; is isomorphic to the algebra of truncated polynomials В[х|/(5^),
and its dimension equals д. |
In the case и = 1 the critical point 0 is called nondegenerate.
It is also logical to set the multiplicity д = 0 for a noncritical point. Indeed, in this case the ideal Гу; contains all elements of
the algebra А and, consequently, Чит Су; = 0.
PROBLEM 7.7. Prove that the definition of a nondegenerate critical point by the condition и = 1 coincides with the definition given earlier: a critical point is called nondegenerate if at it the Hessian
of the function is nonzero.
PROBLEM 7.8. Consider a function {(х): Е" - Е of multiplicity р at
the point 0. Prove the following statements.
1. The germ of the function obtained by multiplying {(т) by an arbitrary
smooth function 9(т), 9(0) = 0, has multiplicity д.
2. The germ of the function Е({(т)): В - В, where Р is an arbitrary smooth
function, Е”(0} = 0, has multiplicity р.
PROBLEM 7.9. Compute the multiplicity of the critical points of functions
of type А„ and О‚, defined in section 4.
PROBLEM 7.10. Compute the multiplicity of the following functions at the origin: { = 12 +92, { = ехр (12 +92), } = х? + хуур, = 12, =? + 2у у, =, Е, Е, уу.
HINT. In many cases, to compute the multiplicity it is convenient
to use the fact that it does not change under a change of coordinates.
Prove this.
For critical points of finite multiplicity (р < со) the following statement holds — a generalization of Morse's lemma, which describes
the case of multiplicity д = 1.
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THEOREM 7.1 (Tougeron!). Let }(х) : В" -+ ® be a smooth function, having at 0 a critical point of multiplicity и. Denote by Т,+1(%) the Taylor polynomial of degree и + 1 of the function }(т) at zero.
Then there exist local coordinates in which the germ of the
function ] coincides with Т/41(т).
Tougeron's theorem substantially facilitates reducing the germ of a function to a normal form at a critical point of finite multiplicity. It suffices to reduce to normal form only the К-jet of the function, where
К = и-+1 (this can be done by means of polynomial changes of variables), without worrying about how the terms of higher degree transform in the process. Indeed, by Tougeron's theorem, the normal form of the function
will coincide with the normal form of its К-jet.
A proof of Tougeron's theorem can be found in
or [39].
PROBLEM 7.11. Let а, ‚с be smooth functions such that
а(0) #0, 65(0) #0, с(0) 20, а(0)с(0) = 572 (0).
Compute the multiplicity of the germ of the function
1(х, у) = а(т,у)=? + 2 т, у)ту? + с(т, уу",
at the point 0, and prove that it has a singularity of type Ад, i.e. it reduces to the normal form +12 + у by a suitable change of coordinates.
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