From: "G. A. Edgar"
Subject: Re: grad f = 0 on a continuous curve
Date: Fri, 26 Mar 1999 15:01:40 -0500
Newsgroups: sci.math.research
Keywords: Functions not constant on a curve of critical points
In article <7dg7cj$p68@aix4.segi.ulg.ac.be>, Pierre-Antoine Absil
[address deleted] wrote:
> Hi! I'm looking for a proof, or a counterexample, for this problem which
> seems to be very basic (but is it?).
> Consider a continuously differentiable function
> f: R^n --> R: x |--> f(x).
> Consider a continuous curve
> r: [0,1] --> R^n: t |--> r(t).
> Suppose that grad f = 0 on the image of r. That is, df/dx_i (r(t)) = 0 for
> i=1,...,n and for all t in [0,1].
> The question is: Is it true that f(r(0)) = f(r(1)) ?
> If r is C1, then the question is very simple. The function
> t |--> f(r(t)) is C1 and d/dt f(r(t)) = Df r'(t) = 0, so f(r(t))=f(r(0)) for
> all t in [0,1].
> But what if r is only continuous?
> I would be delighted if someone could give me the answer, or hints,
> references...
> PA.
The answer is no. Surprising, perhaps. It dates from 1935.
Just 4 pages...
Reference:
H. Whitney, "A function not constant on a connected set of critical
points". Duke Math. J. 1 (1935) 514--517.
--
Gerald A. Edgar edgar@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)
==============================================================================
From: campbell@aero.org (L. Andrew Campbell)
Subject: Re: stationary points of a multivariate polynomial
Date: Tue, 30 Mar 1999 12:39:04 -0800
Newsgroups: sci.math.research
In article <7dm4ia$o14@aix4.segi.ulg.ac.be>,
Pierre-Antoine Absil [address deleted] wrote:
>I have got a question concerning the structure of the set of stationary
>points
>of a multivariate polynomial. The answer is likely to be "yes", but I did
>not
>manage to find a proof. Here it is:
>
> Is it true that the set of stationary points
> of a multivariate polynomial f is the finite union
> of connected sets on each of which f is constant?
>
>By multivariate polynomial, I mean a polynomial function f: R^n --> R.
>For example, f(x1,x2) = x1^2 x2 + x2^2.
>
>By stationary points of f, I mean the points where grad f = 0.
>
>
>As far as the "finite union of connected sets" is concerned, it seems that
>my proposition is true, by a well known theorem of algebraic geometry.
>But what for "f constant"?
>
The answer is indeed yes. A real semialgebraic set (and a fortiori
the algebraic set where grad f = 0) is a finite union of connected
components. A connected semialgebraic set is also semialgebraically
arcwise connected. That is, given two points x and y in a semialgebraic
connected set C, there is a continuous semialgebraic function phi(t),
mapping I into the R^n that contains C, such that phi(0) = x, phi(1) =y,
and all phi(t) are contained in C. Here I is the unit interval [0,1].
This (for the special case of the reals) is the content of Proposition
2.5.11 (p. 37) in [Geometrie Algebrique Reele, by Bochnak, Coste, and Roy].
The set of points of non-differentiability of phi is finite, so the
chain rule can be applied piecewise to deduce that f is constant along
the image of phi.
There are generalizations. The book cited covers the case of real closed
fields, other than the reals, as well; though one must be a bit carefully
with the notion of connectedness (semi-algebraic connectednes, which is
equivalent to topological connectedness in the real case). Prop. 2.5.11
(or rather a sufficient equivalent) can be found in other papers/books
under the heading of curve selection. Finally, the result is true in
so-called o-minimal structures of the appropriate sort (a setting more
general than semialgebraic) - see
[Tame Topology and O-minimal Strucures, by Lou van den Dries].
--
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