Re: classifying degenerate critical points

From: The World Wide Wade (waderameyxiii_at_comcast.remove13.net)
Date: 08/28/04 

Date: Sat, 28 Aug 2004 16:37:35 -0700

In article <a980f606.0408280849.56db2409@posting.google.com>,
 monkey_saddle@yahoo.com (Monkey Saddle) wrote:

> Hi,
>
> I'm looking for some help in classifying degenerate critical points as
> minima, maxima, or saddles. 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?
>
> Here are three examples I've been struggling with, all of which have
> degenerate critical points at (0,0):
>
> f1(x,y) = x^4 + y^4
> f2(x,y) = x^3-3xy^2
> f3(x,y) = (x-y^2)(x-2y^2)
>
> The first is clearly a maximum, the second is the so-called monkey
> saddle, and the third is also a saddle. Historical note: f3 was the
> counter-example used by Peano to disprove a proposition by Lagrange --
> see pg 33 of Han***'s 1917 text "Theory of Maxima and Minima",
> available free online at this long url --
> http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=acl8264.0001.00
> 1
>
> The books I've looked in all seem to just gloss over the issue of
> degenerate critical points, and they state something like, "if the
> Hessian is singular, then the test is indeterminate. The End." So,
> any help in finding a technique to classify degenerate critical points
> would be great, and a reference on this topic would also be wonderful.
> I'm not even sure what topic this would fall under, or where to begin
> looking.
>
> Thanks in advance. . .
> monkey_saddle@yahoo.com

Han*** has Lagrange saying, roughly, "If all the terms of the first and
second dimensions vanish, it is necessary for the existence of a maximum or
minimum that all the terms of the third dimension shall disappear and that
the quantity composed of terms of four dimensions shall be always positive
for the minimum and always negative for the maximum."

Certainly it's necessary for all degree 3 terms to vanish. But an example
like x^6 + y^6 shows that all degree 4 terms can vanish and yet an absolute
minimum occurs at (0,0). I find it hard to believe Lagrange really wrote
the above. On the other hand, the above criterion is certainly *sufficient*
for a local extremum. Hmmmm ...

At any rate, Peano's example says nothing about the "Lagrange proposition"
because the second order terms in (x-y^2)(x-2y^2) do not vanish.

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