OSCILLATION OF ANALYTIC CURVES

The number of zeroes of the restriction of a given polynomial to the t rajectory of a polynomial vector field in (C-n, O) in a neighborhood o f the origin, is bounded in terms of the degrees of the polynomials in volved. In fact, we bound the number of zeroes, in a neighborhood of t he origin, of the restriction to the given analytic curve in (C-n, O) of an analytic function, linearly depending on parameters, through the stabilization time of the sequence of zero subspaces of Taylor coeffi cients of the composed series (which are linear forms in the parameter s). Then a recent result of Gabrielov on multiplicities of the restric tions of polynomials to the trajectories of polynomial vector fields i s used to bound the above stabilization moment.