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](https://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3), 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 $\mathbb P^{2n+1}\_{\mathbb R}$. 



Hypersurfaces without real points
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 $\mathcal U\_{n,d}$.


