Let us consider polynomial contact structures on $\mathbb RP^3$, i.e. contact structures on $\mathbb R^3$ defined by a form $w=Pdx+Qdy+Rdz,\ P,Q,R\in \mathbb R[x,y,z]\ $ in an affine part and then extended to $\mathbb RP^3$, and $ w \wedge dw \ne 0$ everywhere.


One can find all such forms $w$ that $deg P, deg Q, deg R \leq 1$ by direct calculation:


$w=(qy-rz+a)dx+ (pz-qx+b)dy + (rx-py+c)dz,\ a,b,c,p,q,r\in \mathbb R;$ $ap+br+cq \ne 0$.


But I can't do anything for greater degrees. Do you know any criteria for coefficients of $P,Q,R$?


Does anybody know any contact polynomial form with $deg P, deg Q, deg R \geq 2$?


Added: What is the form (I mean form coefficients in $\mathbb R^3\subset \mathbb RP^3$) defining the polynomial contact structure constructed by
plurisubharmonic function $f=x^4+y^4+z^4+t^4$?


Answer: $f=x^4+y^4+z^4+t^4$ is not strictly plurisubharmonic (see on the plane $x=y=0$ on subspace generated by $dx,dy$). So it does not produce a contact structure.



Polynomial contact structures on $RP^3$
**Polynomial distributions on $\mathbb P^n$.** The following works for any field $k$. 
The polynomial $1$-forms defined on $\mathbb A^{n+1}$ which induce distributions on $\mathbb P^n$ are those invariant by homotheties and annihilated by the Euler vector field $R = \sum\_{i=0}^n x\_i \partial\_i$. Explictly these can be written as
$$
\omega = \sum\_{i=0}^n A\_i dx\_i
$$
with $A\_0, \ldots, A\_n$ being homogeneous polynomials of degree $d+1$ satisfying the relation
$$
\sum\_{i=0}^n x\_i A\_i =0 .
$$


In more intrinsic terms $\omega$ is section of $\Omega^1\_{\mathbb P^n}(d+2)$. The integer $d$ appearing above has a nice geometric interpretation when $k=\overline k$ is an algebraically closed field. If we consider a linear inclusion $i: \mathbb P^1 \to \mathbb P^n$ then 
$i^\* \omega$ is a section of $\Omega^1\_{\mathbb P^1}(d+2) \simeq \mathcal O\_{\mathbb P^1}(d)$ and therefore $d$ counts the number of tangencies between the distribution defined by $\omega$ with a generic line. We say that $d$ is the degree of the distribution.


Be careful: the degree of a distribution on $\mathbb P^n$ as defined above does not
coincide with the degree of the coefficients of a polynomial $1$-form defining the same distribution in affine coordinates. Indeed the (maximal) degree of the affine polynomials defining the distribution on $\mathbb A^{n}$ is equal to $d+1$.


**Examples of polynomial contact structures on $\mathbb R\mathbb P^3$ of even degree.** The contact
structures on $\mathbb R^3$ defined by 
$$
(qy−rz+a)dx+(pz−qx+b)dy+(rx−py+c)dz ,
$$
with $ ap+br+cq \neq 0 $, all have degree zero as they can be written in homogenous coordinates $(x:y:z:w) \in \mathbb P^3$ as 
$$
 (qy−rz+aw)dx+(pz−qx+bw)dy+(rx−py+cw)dz + (-ax -by - cz ) dw .
$$
It can also be checked that the induced distributions are all on the $PGL(4,\mathbb R)$-orbit of the one defined by
$$
\omega\_0 = xdy- ydx + zdw- w dz .
$$
Indeed, the action of $\mathrm{PGL}(4,\mathbb C)$ on $\mathbb P H^0 ( \mathbb P^3, \Omega\_{\mathbb P^3}(2))$ has only two orbits. The closed one corresponds to the integrable $1$-forms ( foliations singular along a line ) 
while the open one corresponds to contact structures.



> 
> **Clarification.** The space $\mathbb PH^0(\mathbb P^3, \Omega^1(2))$ can be naturally identified with $\mathbb P ( \bigwedge^2 \mathbb C^4)$. Indeed, the exterior differential is an injective map from linear homogeneous $1$-forms annihilated by Euler's vector field to constant $2$-forms; and the interior product with Euler's vector field sends constant
>  $2$-forms to linear homogeneous $1$-forms annihilated by Euler's vector field. Under these
>  maps the integrable $1$-forms correspond to decomposable $2$-forms. In other words, the foliations in $\mathbb P H^0(\mathbb P^3, \Omega^1(2))$ correspond to the Plucker embedding of the Grasmannian of lines in $\mathbb P^3$ into $\mathbb P (\bigwedge^2 \mathbb C^4)$. 
> 
> 
> 


To produce polynomial contact structures of any even degree $2d$ we have just to multiply $\omega\_0$
by an even homogenous polynomial $P\_{2d} \in \mathbb R[x,y,z,w]$ without non-trivial real solutions and perturb the result in $H^0(\mathbb R \mathbb P^3, \Omega^1(2d+2))$. Since 
$$
(P\_{2d} \omega\_0) \wedge d (P\_{2d} \omega\_0) = P\_{2d}^2 \omega\_0 \wedge d \omega\_0
$$
does not vanish at any point of $\mathbb R \mathbb P^3$, 
we obtain that any section of $ \Omega^1(2d+2) $ in a ~~Zariski~~ sufficiently small
(analytic) neighborhood of $P\_{2d}\omega\_0$ also defines a contact structure. 


**There are no polynomial contact structures of odd degree on $\mathbb R \mathbb P^3$.**
If we have a nowhere zero section of real vector bundle $E$ on a compact manifold $X$ then
the top [Stiefel-Whitney class](http://en.wikipedia.org/wiki/Stiefel-Whitney_class) of $E$ vanishes. From [Euler's sequence](http://en.wikipedia.org/wiki/Euler_sequence)
$$
0 \to \Omega^1\_{\mathbb R \mathbb P^n} \to \mathcal O\_{\mathbb R \mathbb P^n}(-1)^{\oplus n+1} \to \mathcal O\_{\mathbb R \mathbb P^n} \to 0
$$
we can deduce that
$$
w\_n( \Omega^1\_{\mathbb R \mathbb P^n}(d+2) ) = \sum\_{i=0}^n (-1)^i (d+1)^{n-i} \mod 2 .
$$
Notice that the same formula (without the $\mod 2$) counts the number of singularities of
a polynomial distribution over an algebraically closed field if the singularities are isolated.


Specializing to $\mathbb R \mathbb P^3$ we get
$$
w\_3 ( \Omega^1\_{\mathbb R \mathbb P^3}(d+2) ) = \left\lbrace 
\begin{array}
 00 &\text{ if } d \text{ is even} \newline 
 1 &\text{ if } d \text{ is odd}
\end{array}\right. 
$$
and we see that there are no contact distributions of odd degree on $\mathbb R \mathbb P ^3$. 


**Historical remark.** The inexistence result above can be traced back to [Habicht](http://ams.impa.br/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=habicht&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=11&mx-pid=16748) (1948). He dealt with a somewhat different problem which admits an equivalent algebraic formulation. His motivation came from Poincaré-Brower
Theorem about the inexistence of continuous vector fields on the sphere $S^2$. If one looks for homogeneous polynomial vector fields on $\mathbb R^{n+1}$ tangent to the unitary sphere $S^n$ one ends
up with $n+1$ homogeneous polynomials $(f\_0, \ldots, f\_n)$ satisfying $\sum x\_i f\_i=0$. Of course, this is the same as homogeneous polynomial $1$-forms annihilated by Euler's vector field. 


