The computation of the index of a Morse function at a critical point

(1)

THE

COMPUTATION OF THE INDEX

OF

A

MORSE FUNCTION

AT A CRITICAL POINT

TAKISSAKKALIS

Department

of Mathematical Sciences

NewMexicoStateUniversity Las

Cruces, NM

88003

and

IBM

T.J. WatsonResearch Center P.O. Box218

Yorktown Heights,NY 10598

(Received

April 24, 1987 and in revisedform February 12,

1988)

ABSTRACT. A theoretical approachin computingthe index of a Morse func-tion at a critical point on a real non-singular hypersurface V is given. As a consequence the Euler characteristic ofVis computed. Inthe case where the

hypersurfaceispolynomial and compact,aprocedureisgiventhat findsalinear function

g,

whose restrictionglv,isaMorsefunctionon V.

KEY

WORDS AND PHRASES. Morsefunction, critical point, index. 1980 AMS SUBJECT CLASSIFICATION CODES. Primary58, Secondary55. 1. INTRODUCTION.

Let

f(Xl,... ,x,)

be arealC function, andset

v

{(,...,)

e

n"lf(,...,,,)

0}.

Suppose

V is non-singular in R". Furthermore, let

g(xl,.

,x,)

be a real

function whose restriction gig, on

V,

is a Morse function. Then we villfu’st give a theoretical approach of how the Morse index ofgig at a critical point

a can be computed. Using the above data, we can also compute the Euler characteristic,

x(V),

ofV.

(2)

2.

TIIE

BASIC RESULT.

\e first recall some well known results from

Morse

Theory

[3].

For

A a k x k real non-singularsymmetric matrix, we denote by the index of

A, (A),

the number of negative eigenvalues of A. Using the above definition, wc may define the index ofa

Morse

function at a critical point. Let # IV

R

bca real Morsefunction on ar-manifold

W,

andalso let w E W be acritical point of

.

For u,...,

u

localcoordinates onWaroundw we can form the Hessian matrixof# withrespect tou,..., u,.,

H#(u),

II#(u)

o,,o,,

], 1

<_

,

<_

r.

Although the Hessian matrix

H#(u)

dependson the particularcoordintes u,

its index doesnot. We thendefine:

DEFINITION 1. Theindex of at thecriticalpoint w,

i(w)

i(H#(u))

for somecoordinates uaroundw.

Let us now fix some notation. Por

R(z,...,z,)

a real C function, we

OR OR

j=l n.

denoteby

R,

_77,, 1,...,n,

R,

_0,0,,

Letabeacriticalpoint ofgig. Without loss ofgenerality,we mayassume

that

f,,(a)

O. Thenusing the Implicit Function Theorem wemay "solve" the

equation

f(x,...

,xn)

0 for x,,i.e. neara, Vcanbethought asthegraphof Xn

Xn(Xl,...,Xn--1),

and, therefore,x,...,xn-1 are local coordinates forV

neara. Ifwe differentiatethe equation

f(x,... ,x,)

0 twice, and evaluate at

a,veget:

(I)

o=f,.+f,.,,.x,,,+f,,:.x,,+f,m.x,.,,.x,.,,+f,.,.x,,,

,i,j l,...,n-1.

Ata againvehave,

(II)

g,=Af,, i=l,...,n,

AER.

Now

g[

vwithrespecttothecoordinatesx1, Xn--1becomes

Q(x

,...,

x,,_

g(xl,...

,x,_l,X,.,(x,...,

x,,-1)).

Tocompute thcrefore

i(a),

itisenoughto cal-culate theHessian matrix

HQ(xl,...

,x,-l),

at a. Wehave

Q,

g,

+

g,.,x,.,, and

Qo

g,.i

+

g,’,

x,.,,

+

g,, x,,

+

g,.,,

x,.,

x,.,,

+

g,.,

x,,,

Substitutingin

III

what

x,,,

isin and taking

II

intoaccount, xveget

1

(h,i

f

h,.,ff,

h,.,fif,

+

h,.,,f,f)

(IV)

Q,,

where h g

A

f,

A

isthe constant in

II,

and1

_

i,j

_

n- 1.

Ve computed the Hessian matrix

gQ(xl,...,x,.,-1)

(Q,)

at a.

But,

unfortunately, this matrix depends on the particular coordinates uscd at the

point a. Letus nowgiveacoordinatefreematrixwhoseindex isrelatedto

i(a)

inalinearmanner.

(3)

0

v

f)

N=

V’f

H(h)

whereH(h)=h,,,

i,j=l,...,n

The following propositionis themain resultin this paper.

PROPOSITION 1. For

a,h,f,N

as above,

N

is a non-singular matrix.

Furthermore,

i(a)

-i(g) 1.

The proof ofProposition 1 will be in stages. First we will state some

generalitiesandthencomeback to theproof.

For Aa n xn real symmetric matrix we associate the real bilinear form

q(x,y)

xtAy.

We say that q isnon-degenerate if

(q(z,y)

0 Vy) => z 0. This isequivalentinsaying that Aisaninvertiblematrix. Since

A

issymmetric thcreexists an invertiblematrix

P

such that

PtAP

is diagonal. Furthermore,

i(d)

i(P’AP)

[2].

Suppose

A

(aij),i,j 1,...,n is a real symmetric matrix. Let

v

(v,..., v,)

R sothat v,

-

0. Consider the following real symmetric

matrix

B,

B=

vt

A

Fore0,e,...,e, theusualbasis of

R’+,

wehave

<

Be0,e0

>

0

< Beo,

e,:> v,, 1,...,n

<

Bei,

e

>

aij, i,j 1,...,n,

where

<

>

denotesdotproduct.

Now introduce a new basis e0,l,...,,-l,e, on

R

’+1 _so _that

i

v,,e, v,e,,, 1,...,n- 1. Withrespect to those coordinates, the bilinear

form

r(x,

y)

xtBy

gets transformedtoonewhosematrix is

ptBp,

where

P

ia

thefollowingmatrix

1 0 0

0 v, 0

P ₀ "’. 0 and

PtBP

becomes

vn

0

0 --V1, --tn_ 1

PtBP

O, 0,...,0, vn

0

F

0

where

F

(70)

i,j 1,...,n

and7ij

v2na,.

--VnV.ain--

v,v,.,a,.,.

+

v,v.a,,.,,

i,j 1,...,n--1. Wethen have:

LEMMA

1. Suppose

F’

(7,j),

i,j 1,...,n- 1, 7ij as above, is

non-singular. Then

B

isalso non-singular andfurthermore

(4)

PROOF. ,Ve observe that

det(PtBP) #

0,sincev,,

#

0, and therefore Bis

non-singular. Let

R

bearealnon-singular

(n-1)

x

(n-1)

matrix,sothat

RtF’R

is diagonal. Since

F’

is non-singular, all of the diagonal elementsof

RtF’R

are non-zero. Let

R’

be thefollowing non-singular matrix

1, O,

O)

0 R

0

0, 0, 1

NowconsiderS

R’t(ptBP)R’,

Shasthe form

S

Let E, bethefollowing

(n

+

1)

x

(n

+

1)

elementarymatrix.

I 0,

0)

E,=

0,

-,’,

1

where

-

appears in the

(i

+

1)

tt’ _column_for ₁ _n- _1. _{Observe that}

"h

each

Ei

is invertible. Furthermore, a computation shows that Vl’-I E_11;=1 S.

n--1

]-It=l

E, has thefollowingform

O, O, O, vn

0 "[1 0

S’=

"- where b

bn

0 ")’n-- 0

v, 0 0 b

n--I

=1%

On the other hand,

i(S’)

i(S).

To complete the proofof the lemma it is

enoughtoshow that

(v)

i(S’)

#{i’s[%

<

0)

+

1.

To achieve that we look at the

det(S’- AI)

A)(b

A)

,,-1

rt--l(,i

))

H,’--I

n--1

("/i

A)’(2-b-v)

Buttherealrootsof

A

-bA-v

Vn

t=l

eextly

wo,

onepositiveandonenegative.

I

PROOF OF PROPOSITION 1. With thesamenotationandthesamechange ofcoordinates,wetake

B

tobe

N,

then

F

becomes A.

AndnowLemma1 says

i(N)

i(Q)

+

1

i(a)

+

1. |

To compute the index,

i(N),

ofN wefirst look at the negative zeros of

D(z)

det(U

xI).

To determine the number ofnegative zeros of

D(z)

we

can usethe following argument: Let

do

9.c.d(D,D’), dl

9.c.d(do,

do),...,

(5)

d,+

dd._A _0, _k-_1. _Then_we_observe that

1 dx’

1

.D=

D

do

dl

dk---i=o’’"k

-

,j=O k,d_ =D.

d

do

dl

d

Furthermore, we note that each has simple roots and (negative rootsof

D)

=0

(negativeroots of

).

Finally we can use Sturm’s Theorem to decide the number of negativezerosofeach

8j

[1].

IfN happens to be nice, in the sense that no more than tvo consecu-tiveprincipal minors ofN are singular, then

i(N)

variation of sign of the

determinants of itsprincipalminors

[1].

The computation of the Eulercharacteristic,

x(V),

ofV does not require the computation of theindex of

N,

but rather the signof its determinant. We hv

x(V)

(--1)

’(p)-critical of

But

(-1)

i() _sign

de(M)(p)=-

_sign

de(N)(p).

Therefore,

x(V)=-

Ep

sign

det(N)(p).

.

A

THEORETICAL

Fromnowonsuppose that

f

iscompact and non-singularin

Le

{’RH

,

isline,

0).

Then canbe identified with

H-

{0}.

Wehave:

LEMMA

2. Foralmost all elementsgof

:,

g[v

is aMorse functiononV. PROOF. Letr/ V ,n-1 be theGauss

Map.

ThenfromSard’s Theorem

weget that the set of critical values of

r/has

measure zero in S

"-.

For tE

,

g[y

isnot a

Morse

functionon

V

ifandonlyif isacriticalvalue of71

[4].

|

DEFINITION. For

f(xl,...,xn)

a real polynomialofdegree d,d

_>

1, the borderedHessian,

BH(f),

of

f

isthefollowing

(n +:1)

x

(n

+

1)

real symmetric

matrix

(

0

Vf

)

where

H(f)is

theHessian matrixof

BH(f)=

Vt

f

H(f)

f"

Letnow a

(al,... ,a,,)

E N"

{0},

andconsiderthelinear function

g(z) =<

a,x >. Let

L

t[v,

and pacritical point ofL. We may suppose

f,.,(p)

7

O.

Then ifh =/-

A

f,wheregi Af,, 1,...,n. p isanon-degenerate critical pointofLif andonlyif thefollowingmatrixNis non-singular

N=

V’f

H(h)

V’f-AH(f)

[4].

(6)

COROLLARY 2. Fora,

g,

Las above, Lis aMorsefunctionon

"

ifasdonly

if does notbelong to the imageoftheset A

(det BH(f)

0)

NV ,ruder the Gaussmapr/.

REFERENCES

1.

GANTMACHER,

F. R. TheTheoryofMatrices, Vols. I,

II,

Chelsea, 1960. 2.

MACLANE, S.,

BIRI,[HOFF, G. Algebr..a, Macmillan, 1970.

3.

MILNOR,

J. Morse Theory, AnnalsofMath. Studies, Princeton University

Press,

#

51.

4.

SAKI,;ALIS,

T.

"An

Algorithmic ApplicationofMorse Theory to Real