Let Р(т1,..., ть): В" - В be a smooth function having the origin
of coordinates 0 as a critical point. Denote by Н the Hessian matrix of the function РГ at the point 0 (see Remark 1.4). Recall that
a critical point 0 is called nondegenerate if the rank of the matrix
Н equals п (i.e., the maximum possible). We proved earlier the Morse lemma: in suitable local coordinates the germ of the function at a
nondegenerate critical point reduces to the quadratic form (1.8). In this section we study degenerate critical points. Our goal is to reduce the germs of functions at such points to the simplest possible normal form.
4.1. Corank and codimension of degeneracy
The «degree of degeneracy» of a critical point is naturally measured by how much the rank of the matrix Н at it is smaller than the number п. The corresponding number is called the corank. Namely, a critical point at
which rk Н = r, is called a point of corank п — r. It is also often said that the rank of the Hessian at the point 0 «drops by п — r».
Clearly, the corank can also be defined in a more general situation,
when Н is an arbitrary п × п matrix, not necessarily symmetric. Moreover, the notion of corank makes sense for an arbitrary
т × п matrix as well. In this case the corank of a matrix A is usually called
the quantity min{т, п} — r, where r = rk A, showing how much the rank
of the given matrix A differs from the maximum possible rank for a
matrix of the given size. Moreover, it is sometimes necessary to use58
both quantities п—r and т—r; they too are called coranks of the given
matrix (see, for example, Problem 4.1).
In singularity theory the notion of «codimension of degeneracy» plays an important role. This notion applies to objects of various
classes (functions, mappings, vector fields, differential equations, etc.) and makes it possible to determine how typical, or conversely, how exotic a degeneracy of a given type is
within that class. Its essence is as follows.
Each type of degeneracy is defined by some (finite or infinite) number of distinct conditions of the type of equalities and/or inequalities. The codimension of a degeneracy (notation: codim) is the number
of distinct equality-type conditions defining a degeneracy of the given type (inequalities are not counted here). The larger the codimension of a degeneracy, the more rarely a degeneracy of this type occurs
among objects in general position (typical objects of the class under consideration). Namely, if М ⊂ В" is a subset consisting of degenerate points of a certain type, then the equality holds
dim М + codim М = п. (4.1)
Let us illustrate this with the example of critical points of smooth functions of п variables. A nondegenerate critical point of the function Ё(т1,..., тп) is defined by п equalities
Fт1 =0, ..., Fтп = 0 (4.2)
and the inequality det Н ≠ 0; consequently, the set of all critical
points has codimension п. A degenerate critical point is defined by п + 1 equality-type conditions: to equalities (4.2) is added
the further condition det Н = 0.
The set of all critical points and the set of nondegenerate
critical points have codimension п, and, according to formula (4.1),
dimension п — п = 0. This means that critical points of a function
in general position are nondegenerate and isolated. The set of degenerate critical points has codimension п+1 and, consequently,
dimension п — (п + 1) = —1. A negative dimension means
that degenerate critical points are unstable with respect to arbitrarily
small perturbations of the function and, consequently, do not occur for functions in
general position.
Of course, it is not hard to give examples of functions with degenerate critical points. For example, the set of critical points of the
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function Р(т1,т2) = f(т1 + т2) is defined by a single equality f’ = 0.
For each value и such that f’(и) = 0, in the plane (т1,т2)
there is a line т1 + т2 = и consisting entirely of critical points.
Finally, for constant functions, all points of the plane whatsoever are critical. There is no contradiction here: in the
examples given we used functions possessing some specific property or other that is destroyed under an arbitrarily small perturbation. This is precisely what it means to say that the functions we have chosen do not belong
to the case of general position.
PROBLEM 4.1. Prove that in the space of т × п matrices the set of matrices of rank r has codimension
codim = (т—r)(п—r), 0 ≤ r ≤ min{п,т}. (4.3)
This formula is called the «corank product formula»; its proof can be found in the book […] (Section 2.2, «Stratification of
linear operators»). But it is better to prove it yourself.
EXAMPLE 4.1. In the six-dimensional space of 2 × 3 matrices
А — (а11 а12 а13)
(а21 а22 а23)
consider the sets М0 = {А : rk А = 0} and М1 = {А : rk А = 1}. The corank product formula (4.3) gives for them the following result:
codim М0 = (2 — 0)(3 — 0) = 6; codim М1 = (2 — 1)(3 — 1) = 2. The first
result is obvious: the condition rk А = 0 means that all
elements of the matrix А vanish, and this gives six independent equations.
The second result needs a comment. The condition rk А = 1 means that all three second-order minors of the matrix А vanish, and this gives not two, but
three equations. In fact, of these three equations only
two are independent, and the third is a consequence of the other two. To
see this, denote by Аi the second-order minor obtained by deleting the i-th column of the matrix А. Then the
linear relations hold
а11 А1 — а12А2 + а13А3 = 0, а21А1 — а22А2 + а23А3 = 0,
the validity of which can be verified in various ways
(for instance, one may recall the formula for the vector product).
Since a matrix А ∈ М1 has at least one element different from zero, at
least one of these equalities allows one (locally) to express
at least one minor Аi as a linear combination of the other two.
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PROBLEM 4.2. Prove that in the space of germs of smooth
functions of п variables, the set of germs having a critical
point of corank k = п — r has codimension
codim = п + k(k+1)/2, 0 ≤ k ≤ п. (4.4)
Hint. The first term п obviously comes from the
conditions (4.2). It remains to show that in the space of all symmetric matrices of order п × п the set of matrices of corank k has codimension k(k+1)/2. The proof of this statement proceeds according to a
scheme analogous to the proof of the corank product formula
(4.3). The difference between formulas (4.3) and (4.4) lies only in the fact that in the first
case one takes arbitrary т × п matrices, whereas in the second one takes symmetric
п × п matrices.
Despite the fact that critical points of functions Ё(т1,..., тп) of codimension п + k are unstable and, consequently, do not occur for
functions in general position, their study is by no means devoid of meaning. Such critical points occur for families of functions depending on k parameters. Within the class of families of functions depending on k
(or more) parameters, they are stable: if by means of a
small perturbation one destroys such a point in one function of the given family, a critical point of the type under consideration will appear
in another function of the family. For example, from formula (4.4) it follows
that critical points of corank 1 (2) have codimension п + 1 (respectively, п + 3). Therefore they are stable in families of functions
depending on not fewer than one (respectively, three) real parameters.
EXAMPLE 4.2. Consider the family of functions
Ё(т, р) — т³ — тр, (4.5)
depending on the real parameter р. For each fixed value of р the function Ё has exactly one critical point т = р,
which is nondegenerate if and only if р ≠ 0. The degenerate critical point of the function Р(т,0) can be turned into a nondegenerate one,
for example, by means of the perturbation Ё(т, 0) + εт with arbitrarily small
|ε| ≠ 0. If, however, we apply this perturbation to the whole family, i.e.,
consider the family Р(т,р,ε) = т³ — тр + εт, then the degenerate critical
point т = 0 appears at the value р = ε2/3… Thus,
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by our perturbation we have merely shifted the degenerate critical
point from one value of the parameter to another.
It is easy to see that the same holds for any other small
perturbation of the family (4.5). The equations Ёт(т,р) = 0 and Ётр(т,р) = 0
define, in the plane (т, р), the lines т = р and т = 0, transversally
intersecting at zero. Consequently, under a small perturbation of the family (4.5) the equations Ёт(т,р) = 0 and Ётр(т,р) = 0 define, in
general, certain curves, transversally intersecting at some point close to zero — and this is precisely the new degenerate critical point.
Representation (1.9) from Problem 1.3 shows that in order to
obtain normal forms of germs of functions of п variables at degenerate critical points of corank п — r, it suffices to obtain
normal forms of germs of 2-flat functions of п — r variables:
Ё(т1,...,тп—r,у1,...,уr) = ±т1² ±...±тп—r² + f(у1,...,уr), (4.6)
where f = f(у) is a 2-flat function at 0 of п — r variables.
In this section we study degenerate critical points of
functions of corank 1 and 2. For convenience we shall always assume that the critical point under consideration is the origin of coordinates 0
and that the value of the function at the point 0 is zero. The general case, obviously,
is obtained from this one by a shift in the space of arguments and by adding to the right-hand sides of (4.6), (4.7), (4.16) the constant Ё(0).
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