Abstract
We deal with polynomial vector fields of the form ∑dk=1Pk(x1, …, xd)∂/∂xk with d ⩾ 2. Let mk be the degree of Pk. 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have