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Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify that $H$ is smooth.

Q1: Is it possible to compute the number of (topological) connected components of $H$ in term of the $f$?

Q2: How does one compute a cellular decomposition of a connected component of H ?

Q3: What about Q1 and Q2 in the case where one deals with a quadric of signature $(p,q)$?

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    $\begingroup$ A quadric should be expressible as a homogeneous space of $\text{SO}(p, q)$ (or so) and this should make it straightforward to compute e.g. its de Rham cohomology. $\endgroup$
    – Qiaochu Yuan
    May 31, 2014 at 2:52
  • $\begingroup$ Re: the number of connected components, here's a family of examples that shows that there is some possibility for interesting behavior here. If $f$ is the polynomial $y \Delta$ where $\Delta$ is the discriminant of a polynomial of degree $n$ as a function of its coefficients, then $H$ is the complement of the discriminant locus and its number of connected components is the number of involutions in $S_n$ (the involution being the action of complex conjugation on the complex roots). $\endgroup$
    – Qiaochu Yuan
    Jun 11, 2014 at 3:52
  • $\begingroup$ Hi @Qiaochu; this is an interesting observation. $\endgroup$
    – Hugo Chapdelaine
    Jun 11, 2014 at 20:55
  • $\begingroup$ Ah, sorry, that should read "number of conjugacy classes of involutions." $\endgroup$
    – Qiaochu Yuan
    Jun 13, 2014 at 17:17

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There is considerable work on the combinatorics of "arrangements of surfaces," e.g.,

  DanHalperin
Theorem 21.1.4 there says that for a collection of $n$ algebraic surfaces in $\mathbb{R}^d$, under various reasonable general-position assumptions, the combinatorial complexity of the arrangement is $\Theta(n^d)$. So the number of topological connected components is $\Theta(n^d)$.

For how to computationally construct the arrangement, see

Efi Fogel, Dan Halperin, Ron Wein. CGAL Arrangements and Their Applications: A Step-by-Step Guide. Springer, 2012. (Springer link)

from which Fig.2.1 below is taken.
      Fig2.1

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