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]. -- zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz | L. Andrew Campbell internet: campbell@aero.org | | M1-102 PO Box 92957 organization: The Aerospace Corp. | | Los Angeles CA 90009 telephone: (310) 336-8642 |