In this post about a non-linear system of equations solutions were sougth. It turned out that there were no non-trivial solutions for the given numerical values. However, with different numerical values there should be solutions.

In analogy to the fundamental theorem of algebra (which clearly states the number of roots),
I wonder if Maple provides commands that can estimate an upper bound of roots/solutions for a system of multivariate equations by analysing the structure of the system of equations rather than attempting to solve it.

I am not sure if rules, methods or theorems exist at all for multivariate problems. Maybe it is a stupid question but for me this was non-trivial for the case given above.