## Trajectories of polynomial vector fields

**Definition**

Let be a finite piece of a real analytic curve. Its intersection complexity (or intersection degree) is the maximal number of isolated intersections of with real affine hyperplanes,

The goal of this lecture is to explain the following result (D. Novikov, S.Y., 1999) which claims that a sufficiently small piece of a nonsingular trajectory of a polynomial vector field has a finite intersection degree bounded in terms of the dimension and the degree of the field.

More specifically, we consider the polynomial vector field associated with the system of polynomial differential equations

.

Denote its integral trajectory passing through an arbitrary point by .

**Theorem**

For any a sufficiently small piece of has the intersection degree not exceeding .

**References**.

- D. Novikov and S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 563–609, MR1697373 (2001h:32054)
- S. Yakovenko, Quantitative theory of ordinary dierential equations and the tangential Hilbert 16th problem, On finiteness in differential equations and Diophantine geometry, CRM Monogr. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2005, pp. 41–109. MR2180125 (2006g:34062)

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