Polynomial contact structures on $RP^3$ 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.
 A: A plurisubharmonic function $\phi$ on ${\Bbb C}^2$ defines a contact structure on its level set. If $\phi$ is homogeneous, this level set has a real algebraic projection to ${\Bbb R} P^3$, which is a double covering and compatible with the contact structure.
To find such $\phi$, take any homogeneous convex polynomial function on ${\Bbb R}^4$, it would be automatically plurisubharmonic. 
A: Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure
R = QQ[x,y,z,t]
degf=4
f=x^degf+y^degf+z^degf+t^degf + 3*x^2*z^2 + x^2*t^2
--f2 = x^degf+y^degf+z^degf+t^degf
nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t
cdx = degffdiff(y,f)-nR*diff(x,f)
cdy = -degffdiff(x,f)-nR*diff(y,f)
cdz = degffdiff(t,f)-nR*diff(z,f)
cdt = -degffdiff(z,f)-nR*diff(t,f)
cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))
re = sub(cxyz,t=>1) 
factor re 

So, $f2=x^4+y^4+z^4+t^4\ $ (in the above notation) does NOT produce contact structure. It is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why the induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )
A: Concerning your question about $f=x^4+y^4+z^4+t^4$. It indeed induces a polynomial distribution on $\mathbb RP^3$. You should consider a map from the unit sphere to $f^{-1}(1)$ and compute a pull-back of the form given by $df\circ J|_{x^4+y^4+z^4+t^4=1}$. It will become polynomial if you multiply it by an appropriate degree of $f$. I have computed the $dx$ coefficient of the resulting form, but it doesn't look very enlightening to me: $x^6y + y^3(x^4+y^4+z^4+t^4) +x^3t^3z - y^4x^3-x^3z^3t$.
A: There is a very good article "Complex contact threefolds and their contact curves" of Yun-Gang Ye where on can find a classification of complex contact structures on threefolds
A: 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 of $E$ vanishes. From Euler's 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 (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. 
A: I will try to clarify below the answers of  Misha Verbitsky and Max Karev by  reformulating then in the language of my other answer. Contrary to what I have wrote before
in the comments of the main post, the distribution considered by Verbitsky is indeed polynomial. 
Let $f(x,y,z,t)$ be a homogeneous polynomial and $H= f^{-1}(1)$. Consider the restriction
of the $1$-form 
$$
\eta = \frac{\partial f}{\partial y} dx - \frac{\partial f}{\partial x} dy + \frac{\partial f}{\partial t} dz -  \frac{\partial f}{\partial z} dt $$ 
at $H$. We want to extend  the distribution defined on $H$ by it  to the whole $\mathbb R^4$ in such a way that the result is invariant by homotheties.
Since $f$ does not vanish on $H$ we can multiply $\eta$ by $f$ and we will still get the
same distribution on $H$. Similarly, since $df$ vanishes identically when restricted to $H$ we can 
sum multiples of $df$ to $\eta$ without changing the distribution on $H$. Therefore the sought homogeneous $1$-form is 
$$
\omega = \deg(f) f  \eta - (\eta(R)) df ,
$$
where $R$ is Euler's vector field. Notice that $\omega$ defines a section of $\Omega_{\mathbb P^3}(2d)$. 
Its restriction to $H$ defines the very same distribution as $\eta$, and if $Z$ stands for 
the divisorial components of its zero set then $\omega$  defines a polynomial  distribution on $\mathbb P^3_{\mathbb R}$ of degree $2\deg(f) - 2 - \deg(Z)$. 
