Trajectories of polynomial vector fields and ascending chains of polynomial ideals

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector held in R-n and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives.