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Let $n, d$ be positive integers. I am interested in the open subset $\mathcal U_{n,d} \subset \mathbb P H^0 ( \mathbb P^n_{\mathbb R}, \mathcal O_{\mathbb P^n_{\mathbb R}}(d))$ corresponding to sections without real zeros. When $d$ is odd and $n$ arbitrary, the set $\mathcal U_{n,d}$ is empty since (homeogeneous) polynomials of odd degree always have (non-trivial) real roots.

Question. Suppose $d$ is even. How many connected components $\mathcal U_{n,d}$ has ? Do we know anything about the Betti numbers of $\mathcal U_{n,d}$ ?

Indeed, motivated by this other question, I am trying to figure out if it makes sense to ask for the number of connected components of the space of polynomial contact distributions of even degree on $\mathbb P^{2n+1}_{\mathbb R}$.

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up vote 2 down vote accepted

If $s$ is a non-zero section whose image lies in $\mathcal U_{n,d}$, then it has constant sign on $V^\ast:=\mathbb R^{n+1}\setminus\{0\}$ and after possibly multiplying by $-1$ we may assume that $s$ is strictly positive on $V^\ast$. The strictly positive $s$ form an open convex cone $C$ (we do not assume that $0$ belongs to a cone) and is hence contractible when non-empty which this one is when $d$ is even. As $C\to\mathcal U_{n,d}$ is a fibration with fibres $\mathbb R_+$ so is $\mathcal U_{n,d}$.

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@Torsten: the point is not that $C$ is a cone, but that it is convex. –  Laurent Moret-Bailly Mar 26 '11 at 8:39
    
Cones in my definition are always convex $u,v\in C,\lambda,\mu\in\mathbb R_+\implies \lambda u+\mu v\in C$. –  Torsten Ekedahl Mar 26 '11 at 8:46
    
Thanks! Simpler than I expected. –  jvp Mar 26 '11 at 11:12
    
Of course what I call cones should really be called convex cones but it seems that I acquired the habit of calling them cones from toric geometry. –  Torsten Ekedahl Mar 26 '11 at 11:47
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