On the invariant hyperplanes for d-dimensional polynomial vector fields

We deal with polynomial vector fields of the form ∑dk≤1Pk(x1, ..., xd)∂/ ∂xk with d ≥ 2. Let mk be the degree of P k. We call (m1, ..., md) the degree of . We provide the best upper bounds for the polynomial vector field in the function of its degree (m1, ..., md) of (1) the maximal number of invariant hyperplanes, (2) the maximal number of parallel invariant hyperplanes, and (3) the maximal number of invariant hyperplanes that pass through a single point. Moreover, if mi ≤ m, i ≤ 1, ..., d, we show that these best upper bounds are reached taking into account the multiplicity of the invariant hyperplanes. © 2007 IOP Publishing Ltd.