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][1], 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 $\mathbb P^{2n+1}_{\mathbb R}$. 





  [1]: http://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3