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Zbl 0947.37008
Novikov, Dmitri; Yakovenko, Sergei
Trajectories of polynomial vector fields and ascending chains of polynomial ideals. (English)
[J] Ann. Inst. Fourier 49, No.2, 563-609 (1999). ISSN 0373-0956; ISSN 1777-5310/e

Summary: We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in ${\bbfR}^n$ 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.

MSC 2000:: *37C10 Vector fields, flows, ordinary differential equations
37C45 Dimension theory of dynamical systems
37F10 Polynomials; rational maps; entire and meromorphic functions

Keywords: chains of polynomial ideals; intersections; integral curves; polynomial vector field

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