Last time I considered the problems related to ‘existence of random zeros’. And for No.5 problem, we can find a solution.
The governing equation:
and . Here is a projection operator onto the xy plane.
Our solution is , here , since is analytic, then .
Thus I only have to look at case, for high dimension spaces, we just project the points onto a lower dimensional one.
For No.4, we just need to consider the solution to No.3
I do not think the randomness can be achieved for No.1 and No.2, but proof needs more work to do.
- Existence of random zeros::Problems (zym8903.wordpress.com)