Oscillation of analytic curves

Abstract:

The number of zeroes of the restriction of a given polynomial to the trajectory of a polynomial vector field in $(\mathbb {C}^n,0)$, in a neighborhood of the origin, is bounded in terms of the degrees of the polynomials involved. In fact, we bound the number of zeroes, in a neighborhood of the origin, of the restriction to the given analytic curve in $(\mathbb {C}^n,0)$ of an analytic function, linearly depending on parameters, through the stabilization time of the sequence of zero subspaces of Taylor coefficients of the composed series (which are linear forms in the parameters). Then a recent result of Gabrielov on multiplicities of the restrictions of polynomials to the trajectories of polynomial vector fields is used to bound the above stabilization moment.