Consider the differential equation $P(y',y'',y''',y'''')=0$ on $\mathbb R$, where $P(x,y,z,w)$ is the homogeneous polynomial of degree $7$ given by $$ 3x^4yw^2-4x^4z^2w+6x^3y^2zw+24x^2y^4w-12x^3yz^3-29x^2y^3z^2+12y^7. $$ This example was given by Rubel in 1981 (Bulletin of the AMS), and he proved that for any continuous functions $f,g\colon\mathbb R\to\mathbb R$ with $g>0$ there is a solution $y$ of the differential equation satisfying $$ |y(t)-f(t)|\le g(t),\quad \text{for all } t\in\mathbb R. $$ Quite impressive. When one reads the proof one understands that all comes from the particular structure of the equation, but really impressive!
My question is the following:
Is there any polynomial of smaller degree leading to the same property?
A perhaps more ambitious question would be: what is the smallest degree of a polynomial having this property? In fact one could also vary the number of variables of the polynomial.