What is meant by positive root of $x^3-x^3-1$? I am a bit confused. I think there must be a mistake.
In a text I read:


The entropy is $2\ln p$, where $$p=\frac{1}{3}\left(\sqrt[3]{\frac{29+9\sqrt{31/3}}{2}}+\sqrt[3]{\frac{29-9\sqrt{31/3}}{2}}+1\right)$$ is the positive root of $x^3-x^3-1$.



But $x^3-x^3-1=-1$... I think that must be a typo? From what is $p$ the positive root?

There is no positive root, or any root at all, of $x^3-x^3-1=0$, since, as you noted, $x^3-x^3-1=-1$ for all $x$, and there is no $x$ that satisfies $-1=0$. This is definitely a typo.

The positive root of $$x^3-x^2-1=0$$
is exactly 
$$\frac{1}{3}\left(\sqrt[3]{\frac{29+9\sqrt{31/3}}{2}}+\sqrt[3]{\frac{29-9\sqrt{31/3}}{2}}+1\right)$$
so that is almost certainly the intended polynomial. 
