Meandering of trajectories of polynomial vector fields in the affine n-space

Abstract

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R n and an affine hyperplane. 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. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated.

Trajectories of polynomial vector fields and ascending chains of polynomial ideals

“…Below we give a complete demonstration of the results announced in [22], focusing more on the issue of chains of polynomial ideals and algebraic TRAJECTORIES OF POLYNOMIAL VECTOR FIELDS 565 varieties, that plays the key role in the proof. Besides, we improved slightly the estimates and simplified the proof in several instances.…”

“…Still this does not give an explicit answer for the global number of intersections. This paper was preceded by the conference paper [22], an extended abstract in which the main result was announced and the principal ideas of the construction have been already exposed together with motivations, but the long technical proof of the main (algebraic) Theorem 4, the cornerstone of the whole construction, was barely indicated.…”

“…Besides, we improved slightly the estimates and simplified the proof in several instances. However, for the sake of readability some parts of the announcement [22] had to be reproduced below in an abridged form.…”

Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions

“…and algebraic hypersurfaces given by {P = 0} where P = P (t, x) is a polynomial of degree d, the following theorem, proved in [3] (see also [2]), gives a bound for the number of isolated intersections in case the magnitude of the domain of the solution and the amplitude of the solution are controlled by the height of the polynomial system, that is, the number max{|v ikα | : As mentioned in [3], Theorem 2 is nontrivial even for linear systemṡ…”

“…Probably, the simplest result in this spirit is the following theorem for the linear ordinary differential equation An analog of this result for a system of ordinary differential equations, viewed as a vector field in space, would concern the number of isolated intersections between integral trajectories of the vector field and hyperplanes (or, more generally, hypersurfaces). For polynomial systems of degree d on R n of the forṁand algebraic hypersurfaces given by {P = 0} where P = P (t, x) is a polynomial of degree d, the following theorem, proved in [3] (see also [2]), gives a bound for the number of isolated intersections in case the magnitude of the domain of the solution and the amplitude of the solution are controlled by the height of the polynomial system, that is, the number max{|v ikα | : k + |α| ≤ d, i = 1, . .…”