  It is known that homogeneous polynomials on Banach spaces cannot be
hypercyclic, but there are examples of hypercyclic homogeneous polynomials on
some non-normable Fr\'echet spaces. We show the existence of hypercyclic
polynomials on $H(\mathbb C)$, by exhibiting a concrete polynomial which is
also the first example of a frequently hypercyclic homogeneous polynomial on
any $F$-space.
  We prove that the homogeneous polynomial on $ H(\mathbb C)$ defined as the
product of a translation operator and the evaluation at 0 is mixing, frequently
hypercyclic and chaotic. We prove, in contrast, that some natural related
polynomials fail to be hypercyclic.
