The density of prime divisors in the arithmetic dynamics of quadratic polynomials

By Rafe Jones. 2006

137 citations recorded by the source corpus.

Let f ∈ ℤ[x], and consider the recurrence given by an = f(an − 1), with a0 ∈ ℤ. Denote by P(f, a0) the set of prime divisors of this recurrence, that is, the set of primes dividing at least one non‐zero term, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a0)) when f is linear has attracted significant study, although it remains unresolved in full generality. In this paper, we consider the case of f quadratic, where previously D(P(f, a0)) was known only in a few cases. We show that D(P(f, a0)) = 0 regardless of a0 for four infinite families of f, including f = x2 + k, k ∈ ℤ∖{−1}. The proof relies on tools from group theory and probability theory to formulate a sufficient condition for D(P(f, a0)) = 0 in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point have been shown to determine many global dynamical properties of a quadratic polynomial. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc735d79ec13108895e96f541d3df69fd1fde5d5

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Citations in the corpus

Periods of rational maps modulo primes (2011, background): “(The results in [13] are stated for x ∈ Z, but the same proof works for arbitrary x ∈ K .”
A case of the dynamical Mordell–Lang conjecture (2012, methodology): “Using results of Jones [18], we can prove the corresponding result for maps of the form " = ( f1, . . . , fg), without the restriction that fi = f j , if each f j is of the form f j (t) = t2 + c j with c j ∈ Z. Theorem 1.8 Let V ⊂ Ag be a subvariety defined over Q, and let " := ( f1, . . . , fg) act”
The Image of an Arboreal Galois Representation (2009, background): “This set of polynomials is quite a large subset of quadratic f (see [ 9, Theorem 3.5 ]).”
Galois theory of iterated endomorphisms (2007, background): “families of quadratic polynomials [ 15, Section 3 ], but otherwise the question remains open.”
Galois theory of quadratic rational functions (2011, methodology): “In order to prove our main results, we generalize techniques from [8, 14, 18] that treat the case of φ a polynomial.”
ABC implies primitive prime divisors in arithmetic dynamics (2012, background): “Using arguments from [18], one can show Proposition 6.”
The Dynamical Mordell-lang Conjecture (2007, background): “ O be the Φ-orbit of a point (x1,...,xg) ∈ P1 g (Q). Then V(Q) ∩ O is a union of at most ﬁnitely many orbits of the form {Φnk+ℓ(x 1,...,xg)}n≥0 for nonnegative integers kand ℓ. Using results of Jones [Jon08], we can prove the corresponding result for maps of the form Φ = (f1,...,fg), without the res”
Current trends and open problems in arithmetic dynamics (2018, methodology): “Over number fields K, it is known that the Odoni index ιK(f, P ) is finite for certain families of quadratic polynomials [123,227], as well as for various quadratic rational functions [128] and higher-degree polynomials [157], and there are examples over global function fields [111].”
Wreath products and proportions of periodic points (2014, methodology): “of ok, it follows that for all but at most ﬁnitely many primes p, we have #Per(fp) = #Per((σ−1fσ)p). The result now follows immediately from Theorem 6.5. Remark 6.6. We note that the techniques Jones [Jon08] uses to control Gal(((fn(x) − t)/C(t))), for f a post-critically ﬁnite polynomial, are compl”
Primitive divisors in arithmetic dynamics (2007, background): “In some nontrivial dynamical cases, various authors [17, 19, 20] have proven that”
Galois representations from pre-image trees: an arboreal survey (2014, background): “In the case where φ ∈ Z[x] is monic and quadratic, this is Conjecture 1 at the end of Section 4 of [22].”