Consider a set of first-order polynomial differential equations of the form:
$X=(x_1,x_2,...,x_n)$
$\frac{dx_i}{dt} = f_i(x_1,x_2,...,x_n)$
where $f_i$ are polynomials.
There are polynomial root separation theorems such as the one by Rump. In addition, there are results that bound the number of zeroes on an interval for the system of equations that I described such as described here. However, are there any results for root separation for a system of polynomial differential equations? I understand that the initial conditions play a critical role, but we can also include the complexity of the initial conditions by the magnitude of those terms/bits to encode the cofficients/etc.