I watched Boaz Katz' youtube video and read Leo Goldmakher's paper. They both are saying that a commutator does not change a value in a formula with one root of a continuous function $\sqrt{f_1}$. And if we want to get a different value, we need nested radicals. That's why formula Cardano have to have nested radicals. My question is why would commutator change value of nested radicals? Any examples?
My understanding is when the commutator is applied to one radical of a continuous function, arg change will go positive and negative eventually adding up to zero because the two closed paths in the commutator are taking in both directions.
But if total arg change of one radical of a continuous function $\sqrt{f_1}$ is zero and total arg change of another continuous function $f_2$ is also zero, then arg change of their sum is also zero. And then arg change of $\sqrt{f_2+\sqrt{f_1}}$ is also zero when a commutator is applied which does not produce a different value.
It's not the case that the arg change for the sum of two functions is the sum of the arg changes of the individual functions. (Or that the arg change of the sum of two functions each of which has an arg change of zero, must itself be zero.)
A good way of seeing this is to think of a "loop having an arg change of zero" as saying that the origin is outside the loop. While the origin may be outside of the loop for $\sqrt{f_1}$, clearly there can be some point c which is inside that loop, and then taking $f_2 = -c$ the loop for $f2 + \sqrt{f_1}$ will include the origin and have a non-zero arg change. So $\sqrt{f_2+\sqrt{f_1}}$ might not return to the same value, and hence the need for commutators of commutators.