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

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

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

Erased reference request as Ekehdal's answer shows the question is trivial
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Jorge Vitório Pereira
Jorge Vitório Pereira
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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 $\mathbb P^{2n+1}_{\mathbb R}$.

Any reference on the subject will be very welcome.

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

Any reference on the subject will be very welcome.

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

deleted 2 characters in body; edited tags; deleted 11 characters in body; deleted 7 characters in body
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Jorge Vitório Pereira
Jorge Vitório Pereira
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Let $n\ge 1$ and $d \ge 2$$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 when $d$ is even ? Do we know anything about the Betti numbers of $\mathcal U_{n,d}$ ?

Indeed, motivated by this other [question][1]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 $\mathbb P^{2n+1}_{\mathbb R}$.

Any reference on the subject wouldwill be highly appreciatedvery welcome.

[1]: http://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3U_{n,d}$ ?

Let $n\ge 1$ and $d \ge 2$ be 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. How many connected components $\mathcal U_{n,d}$ has when $d$ is even ? 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 $\mathbb P^{2n+1}_{\mathbb R}$.

Any reference on the subject would be highly appreciated.

[1]: http://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3U_{n,d}$ ?

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

Any reference on the subject will be very welcome.

Source Link
Jorge Vitório Pereira
Jorge Vitório Pereira
  • 8.8k
  • 2
  • 38
  • 62