Sergei Yakovenko's blog: on Math and Teaching

Monday, December 29, 2014

Lecture 11 (Mon, Dec 15, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 4:43
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Trajectories of polynomial vector fields

Definition
Let \gamma:(-r,r)\to\mathbb R^n be a finite piece of a real analytic curve. Its intersection complexity (or intersection degree) is the maximal number of isolated intersections of \gamma with real affine hyperplanes,

\displaystyle \deg\gamma=\max_{\varPi\subset\mathbb R^n}\#\{t\in(-r,r):\gamma(t)\in\varPi\}<+\infty.

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

\dot x_i=v_i(x_1,\dots,x_n),\qquad i=1,\dots,n,\quad v_i\in\mathbb R[x_1,\dots,x_n],\ \deg v_i\leqslant d.

Denote its integral trajectory passing through an arbitrary point a\in\mathbb R^n by \gamma_a.

Theorem
For any a\in\mathbb R^n a sufficiently small piece of \gamma_a has the intersection degree not exceeding 2^{2^{O(n^3 d)}}.

References.

  1. 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)
  2. 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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1 Comment »

  1. […] of the (local intersectional) degree of a phase curve of a polynomial vector field, produced in Lecture 11, is based on the length of the ascending chain of polynomial ideals generated by consecutive […]

    Pingback by Lecture 12 (Friday Dec. 19, 2014) | Sergei Yakovenko's blog: on Math and Teaching — Tuesday, December 30, 2014 @ 6:09 | Reply


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