On Stable Quadratic Polynomials

Abstract

Abstract. We recall that a polynomial f (X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f (X) ∈ ‫[ޚ‬X] are stable over ‫.ޑ‬ We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f (X) ∈ ‫[ޚ‬X] can be detected by a finite algorithm; this property is closely related to the stability of f (X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

On Sets of Irreducible Polynomials Closed by Composition

“…In this paper we specialize on irreducibility questions regarding compositional semigroups of polynomials. This kind of question has been addressed in the specific case of semigroups generated by a single quadratic polynomial, see for example in [6,5,11,10], for analogous results related to additive polynomials, see [14,15]. It is worth mentioning that one of these results [11,Lemma 2.5] has been recently used in [16] by the first and the second author of the present paper to prove [12,Conjecture 1.2].…”

Irreducible compositions of degree two polynomials over finite fields have regular structure

“…Figure 2. This shows that the irreducible compositions of f and g are precisely those of the form f (n) , f (n) • g, f (n) • g (2) and f (n) • g (2) • f for n ≥ 0.…”

“…Applying the construction, we get the automaton shown in Figure 3. We see that the irreducible polynomials in C S are exactly x, h, h • g • f (n) for n ≥ 0, and h (2) • k for k ∈ C S arbitrary (possibly the identity). Applying Theorem 3.11, it follows that the set of irreducible polynomials in F 3 [x] that can be written as a nonempty composition of degree 2 polynomials is precisely…”

“…It has been of great interest in recent years the study of irreducible polynomials which can be written as composition of degree two polynomials (see for example [1,2,6,8,9,11,13,14,16,17,18]). Such polynomials are also used in other contexts, see for example Rafe Jones' construction of irreducible polynomials reducible modulo every prime [16] or the proof of [3, Conjecture 1.2] in [5].…”

A note on the stability of trinomials over finite fields

“…We also noted in the introduction that Odoni [11] studied the stability of additive polynomial f (x) = x p − x − 1 over F p . His method is different from the methods of the current paper and those of [1,7]. Using Capelli's lemma he showed that the Galois group of f f (x) over F p is the cyclic group of order p and hence f f (x) is not stable.…”