From: Dave Rusin Subject: Morse theory Date: Thu, 24 Feb 2000 11:48:59 -0600 (CST) To: grubb@math.niu.edu Summary: [missing] OK, here's a quick Morse-theory analysis of your example -- the dog food dish. I'll take that as the graph of f = (x^2+y^2)(x^2+y^2-1). Now, this surface has degenerate critical points, so let me give a small modification: I'll work with f = (x^2+y^2)(x^2+y^2-1) + x/10. Critical points are those with df/dx=df/dy=0. The latter equation factors to imply either x^2+y^2=1/2 or y=0, but the first of those possibilities is incompatible with df/dx=0. Thus the only critical points are those with y=0 and 0 = df/dx = 4x^3 - 2x + 1/10 + a multiple of y. These points are roughly (x,y)=( -.73,0) (.05,0), (.68,0) . At these values of (x,y) we find f = -.32, .0025, -.18 respectively. So there are three critical values of f, i.e. three critical (singular) algebraic curves. The curve f=-.32 consists of the single point (-.73,0). THe curve f = -.18 consists of two nested ovals tangent at (.68,0). The curve f = .0025 is a big oval with the single point (.05,0) in the middle. Now, you know that all the other level curves f=alpha are nonsingular, and together they partition all the points of the plane, no two ever crossing. Morse theory says that the curves are homeomorphic on intervals of alpha's not containing any of the three critical values. So you can see all the possibilities using just, say, alpha=-1, alpha = -1/4, alpha = -1/10, and alpha = 1. These are, respectively, the empty set; a single oval; two nested ovals; a single oval. (The _geometry_ is more complicated than the topology of course. Begin with the three critical curves drawn in the plane. As we vary alpha from -.32 to -.18, the level curves expand from the point (.73,0) to form ever-larger pincer-shaped simple closed curves which expand to the limiting (singular curve f=-.18. For larger values of alpha, the level curves consist of a steadily shrinking oval inside the small part of f=-.18 and a steadily growing oval outside the large part of the same curve, approaching the singular curve f=.0025. Then for yet larger alpha, the small oval is gone and the large oval grows without bound, eventually indistinguishable from the circle at the origin of radius r where r^2(r^2-1) = alpha.) As this example indicates, the topological type of the curves f=alpha can be determined with a finite amount of computation when f is a polynomial, assuming you can plot the finitely many singular curves. I hope this isn't too far from what you were trying to accomplish dave ============================================================================== From: Dave Rusin Subject: Hilbert's 16th problem Date: Thu, 24 Feb 2000 10:53:32 -0600 (CST) To: grubb@math.niu.edu Dan, It seems Hilbert's 16th problem _is_ the "disposition of ovals" question; I'm not entirely sure how it got dressed up as a dynamical-systems question in the first citations we looked at. But try reading review 58 #16684 (or more generally look at the responses to "Anywhere = Hilbert and ovals") and you'll get the sense that the question of how the components of an algebraic curve can be nested is (part of) Hilbert's 16th problem. I don't see a result of the form "anything is possible" in the literature, but there are many papers which seem to say, "such-and-such a nesting pattern is/is not possible for polynomials of degree [some low number]"; if a general result were available, I would imagine that results which look like simple special cases wouldn't be publishable... dave ============================================================================== From: Dave Rusin Subject: Morse theory Date: Thu, 24 Feb 2000 11:48:59 -0600 (CST) To: grubb@math.niu.edu OK, here's a quick Morse-theory analysis of your example -- the dog food dish. I'll take that as the graph of f = (x^2+y^2)(x^2+y^2-1). Now, this surface has degenerate critical points, so let me give a small modification: I'll work with f = (x^2+y^2)(x^2+y^2-1) + x/10. Critical points are those with df/dx=df/dy=0. The latter equation factors to imply either x^2+y^2=1/2 or y=0, but the first of those possibilities is incompatible with df/dx=0. Thus the only critical points are those with y=0 and 0 = df/dx = 4x^3 - 2x + 1/10 + a multiple of y. These points are roughly (x,y)=( -.73,0) (.05,0), (.68,0) . At these values of (x,y) we find f = -.32, .0025, -.18 respectively. So there are three critical values of f, i.e. three critical (singular) algebraic curves. The curve f=-.32 consists of the single point (-.73,0). THe curve f = -.18 consists of two nested ovals tangent at (.68,0). The curve f = .0025 is a big oval with the single point (.05,0) in the middle. Now, you know that all the other level curves f=alpha are nonsingular, and together they partition all the points of the plane, no two ever crossing. Morse theory says that the curves are homeomorphic on intervals of alpha's not containing any of the three critical values. So you can see all the possibilities using just, say, alpha=-1, alpha = -1/4, alpha = -1/10, and alpha = 1. These are, respectively, the empty set; a single oval; two nested ovals; a single oval. (The _geometry_ is more complicated than the topology of course. Begin with the three critical curves drawn in the plane. As we vary alpha from -.32 to -.18, the level curves expand from the point (.73,0) to form ever-larger pincer-shaped simple closed curves which expand to the limiting (singular curve f=-.18. For larger values of alpha, the level curves consist of a steadily shrinking oval inside the small part of f=-.18 and a steadily growing oval outside the large part of the same curve, approaching the singular curve f=.0025. Then for yet larger alpha, the small oval is gone and the large oval grows without bound, eventually indistinguishable from the circle at the origin of radius r where r^2(r^2-1) = alpha.) As this example indicates, the topological type of the curves f=alpha can be determined with a finite amount of computation when f is a polynomial, assuming you can plot the finitely many singular curves. I hope this isn't too far from what you were trying to accomplish dave ============================================================================== From: Daniel Grubb Subject: Re: Morse theory Date: Fri, 25 Feb 2000 07:40:12 -0600 (CST) To: Dave Rusin Thanks for the analysis, it was quite enlightening. I looked at the book you loaned me and it has already given me some insights on the type of problems I am dealing with. I am particularly amused by the characterization of the Euler characteristic in terms of the indices of the critical points. Is there anything known about what happens for degenerate critical points? Is it always possible on a compact manifold to add a 'small' function and make all critical point non-degenerate? What happens for functions that are not C^\infty? It seems that a similar treatment should be possible for 'piecewise linear' functions. --Dan Grubb