Progress in J.F.Ritt's one problem?

In Ritt's classical book"Differential Algebra",he asked the following question(it can be found in page 177,problem 8):

The decomposition problem: determine the number of times which the d.p.in a finite system $\phi$ must be differentied before elimitions will produce finite systems whose manifolds are the component of $\phi$.One would hope to secure a bound which depends on the number of d.p.in $\phi$,their orders and degrees.

Can anyone tell me the progress in this problem?

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