[Paper Review] Polynomial Mappings mod p^n

David L. desJardins, Michael E. Zieve|ArXiv.org|Mar 7, 2001

Advanced Differential Equations and Dynamical Systems9 citations

TL;DR

This paper develops a framework for analyzing the cycle structure of polynomial mappings modulo odd prime powers $p^n$, using Taylor expansion and lifting techniques to classify how cycles evolve from $\mathbb{Z}/p^n\mathbb{Z}$ to $\mathbb{Z}/p^{n+1}\mathbb{Z}$. The key contribution is a complete classification of cycle lifting behavior—growth, splitting, tail growth, and partial splitting—based on the derivative and displacement modulo $p$, enabling fast computation of cycle lengths and construction of polynomials with desired dynamics.

ABSTRACT

Let f(x) be a polynomial with integer coefficients, let n be a positive integer, and let p be an odd prime. Then the mapping x-->f(x) sends Z/p^n into Z/p^n. We study the topological structure of this mapping.

Motivation & Objective

• To understand the topological structure of polynomial mappings $f: \mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$ for odd primes $p$ and positive integers $n$.
• To determine how cycles of $f_n$ lift to cycles of $f_{n+1}$, especially the conditions under which cycles grow, split, or develop tails.
• To provide an algorithm for computing cycle lengths of $f_n$ for all $n$ efficiently, given a polynomial $f$, and to enable construction of polynomials with specified cycle structures modulo $p^n$.
• To extend the results to rational functions $h = f/g$ under the condition that $g$ is coprime to $p$ on relevant cycles, by constructing compatible polynomial approximations modulo $p^{2n}$.

Proposed method

• Use Taylor expansion of the $k$-th iterate $g = f^k$ around a cycle point $x_1$ to derive $g(x_1 + p^n t) \equiv x_1 + p^n b_n + p^n a_n t \pmod{p^{2n}}$, where $a_n = g'(x_1)$ and $b_n = (g(x_1) - x_1)/p^n$.
• Define a map $\Phi: \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$ via $\Phi(t) = b_n + a_n t$, which models the action of $f_{n+1}$ on the lift of a cycle, revealing linear dynamics.
• Classify cycle lifting behavior based on $a_n \mod p$ and $b_n \mod p$: if $a_n \equiv 1$ and $b_n \not\equiv 0$, the cycle grows to length $pk$; if $a_n \equiv 1$ and $b_n \equiv 0$, it splits into $p$ cycles of length $k$; if $a_n \equiv 0$, it grows tails; otherwise, it partially splits.
• Leverage the fact that $a_n$ and $b_n$ are well-defined modulo $p$ up to cycle and representative choice, ensuring invariance of lifting behavior under different representatives.
• Extend results to rational functions $h = f/g$ by constructing a sequence of polynomials $h_n = f \cdot g^{\phi(p^{2n})-1}$ that agree with $h$ modulo $p^{2n}$ when $g(x) \not\equiv 0 \pmod{p}$, ensuring compatibility of $a_i$ and $b_i$ across levels.
• Use the compatibility of $a_i$ and $b_i$ across $n$ to show that cycle lifting behavior for $h$ matches that of the approximating polynomials $h_n$, thus generalizing results to rational functions.

Experimental results

Research questions

• RQ1 How do cycles of a polynomial mapping $f: \mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$ lift to cycles modulo $p^{n+1}$?
• RQ2 What conditions on the derivative $f^{(k)\prime}(x_1)$ and displacement $(f^k(x_1) - x_1)/p^n$ determine whether a cycle grows, splits, or develops tails when lifted to $\mathbb{Z}/p^{n+1}\mathbb{Z}$?
• RQ3 Can the cycle structure of $f_n$ be computed efficiently for all $n$, and is there an algorithm to construct a polynomial with a prescribed cycle structure modulo $p^n$?
• RQ4 To what extent do the results for polynomials extend to rational functions $f/g$ where $g$ is coprime to $p$ on the cycles of interest?
• RQ5 What is the role of the $p$-adic structure in determining the behavior of iterated polynomial mappings modulo $p^n$?

Key findings

• Cycle lifting is governed by the linear map $\Phi(t) = b_n + a_n t$ on $\mathbb{Z}/p\mathbb{Z}$, where $a_n = g'(x_1)$ and $b_n = (g(x_1) - x_1)/p^n$ for $g = f^k$, leading to a complete classification of lifting behavior.
• If $a_n \equiv 1 \pmod{p}$ and $b_n \not\equiv 0 \pmod{p}$, the cycle grows to length $pk$; this is called 'growth' of the cycle.
• If $a_n \equiv 1 \pmod{p}$ and $b_n \equiv 0 \pmod{p}$, the cycle splits into $p$ disjoint cycles of length $k$; this is called 'splitting'.
• If $a_n \equiv 0 \pmod{p}$, the cycle grows tails: one $k$-cycle and the remaining $p-1$ points map into it after $k$ iterations.
• If $a_n \not\equiv 0,1 \pmod{p}$, the cycle partially splits: one $k$-cycle and $(p-1)/d$ cycles of length $kd$, where $d$ is the multiplicative order of $a_n$ modulo $p$.
• The results extend to rational functions $h = f/g$ with $p \nmid g(x)$ on cycles, by constructing compatible polynomial approximations $h_n$ modulo $p^{2n}$, ensuring the same lifting behavior holds.