From: wcw@math.psu.edu (William C Waterhouse) Subject: Re: Connected components of a real variety Date: 29 Jan 2001 19:05:09 GMT Newsgroups: sci.math.research Summary: Hilbert's 16th problem: how many components has a real algebraic variety? In article <94v2m7$11m$1@agate.berkeley.edu>, chernoff@math.berkeley.edu (Paul R. Chernoff) writes: > In article <94u14m$edt$1@nnrp1.deja.com>, wrote: > >If V is an algebraic variety defined in R^n by polynomial equations > >f_i = 0 for i=1,2,...m and each polynomial f_i is of degree <= k, > >is there a known upper bound on the number of connected components of V > >in terms of n,m and k ? >... > Seems to me that the R^2 case is one of Hilbert's problems, and so far as > I know is unsolved. ... The Hilbert Problem asked a somewhat more complicated question. Here it is (in the English translation), from the website 16. Problem of the topology of algebraic curves and surfaces The maximum number of closed and separate branches which a plane algebraic curve of the n-th order can have has been determined by Harnack. There arises the further question as to the relative position of the branches in the plane. As to curves of the 6-th order, I have satisfied myself--by a complicated process, it is true--that of the eleven branches which they can have according to Harnack, by no means all can lie external to one another, but that one branch must exist in whose interior one branch and in whose exterior nine branches lie, or inversely. A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maximum number of sheets which a surface of the 4-th order in three dimensional space can really have. William C. Waterhouse Penn State ============================================================================== From: Torsten Ekedahl Subject: Re: Connected components of a real variety Date: 27 Jan 2001 20:44:14 +0100 Newsgroups: sci.math.research david_maor@my-deja.com writes: > If V is an algebraic variety defined in R^n by polynomial equations > f_i = 0 for i=1,2,...m and each polynomial f_i is of degree <= k, > is there a known upper bound on the number of connected components of V > in terms of n,m and k ? There is a result due to Milnor[1] that gives an upper bound, k(2k-1)^{n-1}, for the sum of the (mod 2) Betti numbers of V, in particular this gives a bound for the zeroth Betti number which equals the number of connected components. Possibly, getting an upper bound for the number of components could be simpler. Footnotes: [1] Milnor, J. On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15 1964 275--280. From greg@math.ucdavis.edu Thu Feb 1 23:49:11 CST 2001 Article: 800 of sci.math.research Path: news1!news.math.niu.edu!husk.cso.niu.edu!vixen.cso.uiuc.edu!not-for-mail X-Coding-System: undecided-unix From: greg@math.ucdavis.edu (Greg Kuperberg) Newsgroups: sci.math.research Subject: Re: Connected components of a real variety Message-ID: <94vhlj$aik$1@manifold.math.ucdavis.edu> References: <94u14m$edt$1@nnrp1.deja.com> Approved: Daniel Grayson, dan@math.uiuc.edu, moderator for sci.math.research Lines: 55 Date: 27 Jan 2001 14:21:39 -0800 NNTP-Posting-Host: 130.126.108.30 X-Complaints-To: abuse@uiuc.edu X-Trace: vixen.cso.uiuc.edu 980639931 130.126.108.30 (Sat, 27 Jan 2001 17:58:51 CST) NNTP-Posting-Date: Sat, 27 Jan 2001 17:58:51 CST Organization: University of Illinois at Urbana-Champaign Xref: news1 sci.math.research:800 In article <94u14m$edt$1@nnrp1.deja.com>, wrote: >If V is an algebraic variety defined in R^n by polynomial equations >f_i = 0 for i=1,2,...m and each polynomial f_i is of degree <= k, >is there a known upper bound on the number of connected components of V >in terms of n,m and k ? There is a famous old paper of Milnor, part of which was done independently by Thom, in which he establishes that the total of the Betti numbers of V is at most k*(2*k-1)^(n-1), independent of m. In particular, there are at most that many components. There is a special case which illustrates the idea. Suppose that V is smooth and defined by a single equation f(x) = 0 of degree k and suppose that every component is bounded. let x_0,...,x_{n-1} be the coordinates and call x_0 "vertical". If you intersect V with the equations df/dx_i = 0 for all i != 0, Bezout's theorem tells you that V admits at most k*(k-1)^(n-1) horizontal tangent hyperplanes, which are the critical points of x_0. If V is in general position relative to the coordinates, x_0 is a Morse function, which means that you can use it to compute homology and that the number of critical points bounds the homology. In particular, with the suppositions that I have made, V has at most k/2*(k-1)^(n-1) components, since x_0 must have at least one local minimum and maximum on each component. The general case of Milnor's theorem follows by carefully removing the suppositions without throwing out the baby with the bath water. For starters he reduces to the single-equation case by taking the equation f_1^2 + .... + f_m^2 = 0. Such reductions may seem crude, but the argument is at least short. All attempts at improvement have required a lot of pain for only small gains. It would be interesting to obtain an absolute bound on the topological complexity of such a variety V, even in the special case that I discussed. For example a bound on the minimum number of simplices needed to triangulate the pair (R^n,V) or the pair (RP^n,closure(V)). References: J. Milnor, Proc. Amer. Math. Soc. {\bf 15} (1964), 275--280; MR {\bf 28} \#4547 R. Thom, in {\it Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse)}, 255--265, Princeton Univ. Press, Princeton, N.J., 1965; MR {\bf 34} \#828 -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ \/ * All the math that's fit to e-print *