[Paper Review] The density of primes in orbits of z^d + c

Spencer Hamblen, Rafe Jones+1 more|arXiv (Cornell University)|Mar 26, 2013

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper establishes that for the polynomial $ f(z) = z^d + c $ over a global field $ K $ containing a primitive $ d $-th root of unity, the set of prime ideals dividing any element in the orbit of $ a_0 \in K $ has Dirichlet density zero under two distinct conditions: (1) when $ c $ is $ v $-adic unit with residue characteristic coprime to $ d $, or (2) when certain iterates of $ f $ factor into irreducibles and their values are not $ d $-th powers in $ K $. The result generalizes prior work on quadratic maps to higher-degree polynomials and uses novel techniques in ramification theory and irreducibility bounds.

ABSTRACT

Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we study the density of prime ideals of K dividing at least one element of the orbit of a_0 under f. The density of such sets for linear polynomials has attracted much study, and the second author has examined several families of quadratic polynomials, but little is known in the higher-degree case. We show that for many choices of d and c this density is zero for all a_0, assuming K contains a primitive dth root of unity. The proof relies on several new results, including some ensuring the number of irreducible factors of the nth iterate of f remains bounded as n grows, and others on the ramification above certain primes in iterated extensions. Together these allow for nearly complete information when K is a global function field or when K=Q(zeta_d).

Motivation & Objective

• To determine the Dirichlet density of prime ideals dividing elements in the forward orbit of $ a_0 \in K $ under the map $ f(z) = z^d + c $, where $ K $ is a global field.
• To extend known results on prime density in orbits from quadratic maps to higher-degree polynomials $ z^d + c $ with $ d > 2 $.
• To establish conditions under which this density is zero, particularly when $ K $ contains a primitive $ d $-th root of unity.
• To analyze the arithmetic structure of iterated polynomials via irreducibility and ramification in local and global extensions.

Proposed method

• Uses a local method based on ramification degrees in $ p $-adic extensions of $ K_v $, the completion of $ K $ at a non-archimedean place $ v $, to prove condition (1).
• Applies a global method extending techniques from Jones (2008) to higher-degree polynomials, analyzing the non-$ d $-th power property of values of irreducible factors of $ f^n(z) $.
• Establishes bounds on the number of irreducible factors of the $ n $-th iterate $ f^n(z) $, showing this number remains uniformly bounded as $ n \to \infty $.
• Employs the product formula and norm estimates in $ \mathbb{Q}(\zeta_d) $ to rule out $ d $-th powers in number fields, especially for $ d $-prime.
• Reduces the problem to showing that certain algebraic integers in $ \mathbb{Q}(\zeta_p) $ are not $ p $-th powers, using congruence arguments modulo $ p $.
• Uses Galois conjugacy and norm comparisons to show that $ f^{n-1}(0) + r\zeta_p^i $ cannot be a $ p $-th power in $ \mathbb{Q}(\zeta_p) $ for $ 0 < i < p $.

Experimental results

Research questions

• RQ1 Under what conditions is the set of prime ideals dividing the orbit of $ a_0 $ under $ f(z) = z^d + c $ sparse in the sense of Dirichlet density zero?
• RQ2 Can the results on prime density for quadratic maps be generalized to higher-degree polynomials $ z^d + c $ with $ d > 2 $?
• RQ3 How does the presence of a primitive $ d $-th root of unity in $ K $ affect the arithmetic structure of iterated polynomial orbits?
• RQ4 What role does the ramification behavior in local fields $ K_v $ play in controlling the density of primes dividing orbit elements?
• RQ5 When is the value $ f^n(0) $ not a $ d $-th power in $ K $, and how does this relate to the irreducibility of iterates?

Key findings

• For $ f(z) = z^d + c $ over a global field $ K $ containing a primitive $ d $-th root of unity, the Dirichlet density of prime ideals dividing any element of the orbit of $ a_0 $ is zero under condition (1): if there exists a non-archimedean place $ v $ with $ |c|_v < 1 $ and residue characteristic coprime to $ d $.
• Under condition (2), if $ f^j(z) $ factors into irreducibles $ g_i(z) $ and none of $ \pm g_i(f^k(0)) $ is a $ d $-th power in $ K $ for $ k \geq 1 $, then the density is zero.
• The number of irreducible factors of the $ n $-th iterate $ f^n(z) $ remains bounded as $ n \to \infty $, a key technical result enabling the density analysis.
• In the case $ K = \mathbb{Q}(\zeta_d) $, the density is zero when $ d $ is prime and $ f^n(0) $ is not a $ d $-th power in $ \mathbb{Z} $, which holds for all $ n \geq 1 $ if $ c $ is not a $ d $-th power in $ \mathbb{Z} $.
• For $ d = p $ odd prime and $ c = r^p $, the values $ f^n(0) + r\zeta_p^i $ are not $ p $-th powers in $ \mathbb{Q}(\zeta_p) $, proven via norm estimates and reduction modulo $ p $.
• The result is nearly complete for global function fields and for $ K = \mathbb{Q}(\zeta_d) $, with density zero established under broad conditions on $ c $.