THE COMPUTATION OF THE INDEX OF A MORSE FUNCTION AT A CRITICAL POINT

A theoretical approach in computing the index of a Morse function at a critical point on a real non-singular hypersurface V is given. As a consequence the Euler characteristic of V is computed. In the case where the hypersurface is polynomial and compact, a procedure is given that finds a linear function ℓ, whose restriction ℓ|V, is a Morse function on V.

Let f(Xl,... ,x,) be a real C function, and set 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 vill fu'st give a theoretical approach of how the Morse index of gig at a critical point a can be computed.Using the above data, we can also compute the Euler characteristic, x(V), of V.
Finally, in the case where f is a polynomial, we will say how we can obtain a polynomial function g, whose restriction glv, has no degenerate critical points on 2. TIIE BASIC RESULT.
\e first recall some well known results from Morse Theory [3].For A a k x k real non-singular symmetric 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 of a Morse function at a critical point.Let # IV R bc a real Morse function on a r-manifold W, and also let w E W be a critical point of .For u,..., u local coordinates on W around w we can form the Hessian matrix of # with respect to u,..., u,., H#(u), II#(u) o,,o,, ], 1 <_ , <_ r.
Although the Hessian matrix H#(u) depends on the particular coordintes u, its index does not.We then define: DEFINITION 1.The index of at the critical point w, i(w) i(H#(u)) for some coordinates u around w.
Let us now fix some notation.Por R(z,...,z,) a real C function, we OR OR j=l n.
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.Let us now give a coordinate free matrix whose index is related to i(a) in a linear manner.
Let a,h,f be as before.Consider the following real (n + 1) x (n + 1) The following proposition is the main result in 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 of Proposition 1 will be in stages.First we will state some generalities and then come back to the proof.
For A a n x n real symmetric matrix we associate the real bilinear form q(x,y) xtAy.We say that q is non-degenerate if (q(z,y) 0 Vy) => z 0. This is equivalent in saying that A is an invertible matrix.Since A is symmetric thcre exists an invertible matrix P such that PtAP is diagonal.Furthermore, i(d) i(P'AP) [2].
PROOF.,Ve observe that det(PtBP) # 0, since v,, # 0, and therefore B is non-singular.Let R be a real non-singular (n-1) x (n-1) matrix, so that RtF'R is diagonal.Since F' is non-singular, all of the diagonal elements of RtF'R are non-zero.Let R' be the following non-singular matrix 1, O, O) 0 R 0 0, 0, 1 Now consider S R't(ptBP)R', S has the form S Let E, be the following (n + 1) x (n + 1) elementary matrix.I 0, 0) E,= 0, -,', 1 whereappears 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 S.

=1%
On the other hand, i(S') i(S).To complete the proof of the lemma it is To achieve that we look at the det(S'- H,'--I ("/i A)'(2-b-v) But the real roots of A -bA-v Vn t=l e extly wo, one positive and one negative.I PROOF OF PROPOSITION 1.With the same notation and the same change of coordinates, we take B to be N, then F becomes A. And now Lemma 1 says i(N) i(Q) + 1 i(a) + 1. | To compute the index, i(N), of N we first look at the negative zeros of D(z) det(U xI).To determine the number of negative zeros of D(z) we can use the following argument: Let do 9.c.d(D,D'), dl 9.c.d(do, do),..., Furthermore, we note that each has simple roots and (negative roots of D) =0 (negative roots of ).Finally we can use Sturm's Theorem to decide the number of negative zeros of each 8j [1].
If N happens to be nice, in the sense that no more than tvo consecu- tive principal minors of N are singular, then i(N) variation of sign of the determinants of its principal minors [1].
The computation of the Euler characteristic, x(V), of V does not require the computation of the index of N, but rather the sign of its determinant.We hv x(V) (--1) '(p)- critical of But (-1) i() sign de(M)(p)=-sign de(N)(p).
. A THEORETICAL From now on suppose that f is compact and non-singular in Le {'RH , isline, 0).Then can be identified with H-{0}.We have: LEMMA 2. For almost all elements g of :, g[v is a Morse function on V.
PROOF.Let r/ V ,n-1 be the Gauss Map.Then from Sard's Theorem we get that the set of critical values of r/has measure zero in S"-.For t E , g[y is not a Morse function on V if and only if is a critical value of 71 [4].| DEFINITION.For f(xl,...,xn) a real polynomial of degree d,d _> 1, the bordered Hessian, BH(f), of f is the following (n +:1) x (n + 1) real symmetric matrix ( 0 Vf ) where H(f)is the Hessian matrix of BH(f)= Vt f H(f) f" Let now a (al,... ,a,,) E N" {0}, and consider the linear function g(z) =< a,x >.Let L t[v, and p a critical point of L. We may suppose f,.,(p) 7 O.
Then if h =/-Af, where gi Af,, 1,..., n. p is a non-degenerate critical point of L if and only if the following matrix N is non-singular N= V'f H(h) V'f-AH(f) [4].