Re: classifying degenerate critical points

From: Lee Rudolph (lrudolph_at_panix.com)
Date: 08/28/04 

Date: 28 Aug 2004 13:00:46 -0400

monkey_saddle@yahoo.com (Monkey Saddle) writes:

>Hi,
>
>I'm looking for some help in classifying degenerate critical points as
>minima, maxima, or saddles.

I suspect this is much harder than you think it is. (I mean,
I know it's very hard, and I suspect you don't think that--yet.)

>Assuming a function f : R^n --> R, the
>critical points are those where the gradient of f is zero. The
>"standard" technique for classifying critical points involves
>examination of the eigenvalues of the Hessian at the critical points.
>This is basically an extension of the Second Derivative Test to the
>multivariate case, and is a consequence of Taylor's theorem. However,
>when the Hessian is not invertible (i.e. when the critical point is
>"degenerate" or "non-Morse"), this test breaks down, much as the
>Second Derivative Test breaks down for, say, f(x)=x^4. For this
>scalar example, we have to examine higher order derivatives to
>classify the critical points. Is there a similar technique for the
>multivariate case?

Even in one variable, if by "higher order derivatives" in the phrase
"examine higher order derivatives" you mean only the values of the
higher order derivatives at the critical point, then you *cannot*
find *necessary and sufficient* conditions in terms of "higher
order derivatives" for a critical point to be, or not to be, a
local extremum: there are, as you know, smooth functions which
are "infinitely flat" at some critical point (i.e., all the
derivatives vanish there), and you can readily convince yourself
that such examples exist which are local minima, others which are
local maxima, and yet others which are neither. Things only get
worse (much worse) in higher dimensions.

Now, if you restrict your attention to *real analytic*, or further
to *real polynomial*, functions, then you may be able to get
somewhere. A good keyword phrase to look for is "finitely
determined singularity" or "finitely determined germ".
Maybe Wall's fairly recent _magnum opus_ on singularities
of real mappings would have pointers in its bibliography.

Lee Rudolph