Bertin Diarra, Djeidi Sylla: Dynatomic polynomials associated with distinguished polynomials, 173-186

Abstract:

Let $K$ be a field and $h \in K[z]$ be a monic polynomial. By analogy with a relation giving cyclotomic polynomials, for any integer $\nu\ge 1$, one defines the $\nu$-th dynatomic polynomial of $h$ to be $\Phi_{ \nu, h}(z) = \prod_{d\vert \nu}(h^{\circ d}(z) -z)^{\mu(\frac{\nu}{d})}.$ These polynomials are related to the search of primitive periodic points of the dynamic system attached to $h$. Their studies had been undertaken on one hand by Vivaldi and Hatjispyros [11], on the other hand by Morton and Patel in [5] which give fundamental properties of them. In this paper we are concerned with the polynomials written in the form $h(z) = z + g(z)$; more particularly when $g$ is a distinguished polynomial with cfficients in the valuation ring of a complete ultrametric valued field $L$ with residue characteristic $p\not = 0$. Let $\overline L$ be the residue field of $L$. For $g$ a distinguished polynomial of degree $q$ a power of $p$, we obtain in $\overline L [z]$ reductions of the polynomials $g_\nu (z) = h^{\circ \nu}(z) - z, \nu \ge 1$ which are additive polynomials with cfficients in $\overline L$ independent of $g$ and a reduction of $\Phi_{ \nu, h}$, if further $\nu$ is a prime number. For a distinguished polynomial of the form $g(z) = a_0 +z^p,$ that is $\vert a_0\vert < 1$, we get that the primitive $3$-periodic points of $h(z) = a_0 +z +z^p$ are the roots of the $3$-th dynatomic polynomial $\Phi_{3, h}$ of $h(z)$. We then study for $L$ equal the field of $p$-adic numbers the $3$-th dynatomic polynomial $\Phi_{3, h}$ and its roots, when $p= 2, 3$. For $p=2$, we apply Schönemann irreducibility criterion. For $p =5$, we use Berlekamp algorithm over the residue field $\mathbb{F}_ 5$ to establish irreducibility of a polynomial linked to reduction modulo $5$ of the $3$-th dynatomic polynomial which will be applied elsewhere.

Key Words: Discrete dynamics, dynatomic polynomials, distinguished polynomials, field extensions.

2010 Mathematics Subject Classification: Primary 11S82; Secondary 37P05, 11F85, 12J25.