On the invariant hyperplanes for<i>d</i>-dimensional polynomial vector fields

Abstract

We deal with polynomial vector fields X of the formd=1 Pk(x1 ,...,x d )∂/∂xk with d 2. Let mk be the degree of Pk. We call (m1 ,...,m d ) the degree of X. We provide the best upper bounds for the polynomial vector field X in the function of its degree (m1 ,...,m d ) 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.