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A variant of Collatz's Conjecture over Binary Polynomials

Luis H. Gallardo, Olivier Rahavandrainy

TL;DR

This work studies a binary polynomial analogue of the Collatz Syracuse process over $\mathbb{F}_2$ by defining parity via $x(x+1)$ and constructing an associated dynamical system on polynomials, yielding sequences of even and odd polynomials. It proves a finite-length result (Theorem collatz0) with an exponential upper bound $r_A \le 2^{\deg(A)-1}$, based on degree convergence arguments and a tail linear system with a circulant matrix whose determinant is nonzero. Complementary computational data for degrees up to $35$ suggest the actual sequence lengths may grow polynomially in the degree and motivate conjectures such as $g(n)=\lfloor n/2\rfloor$ and $r_A\le n(n+1)/2$, along with patterns in special families like $A={M_1}^{n}+1$ where $M_1=x^2+x+1$. The results advance understanding of polynomial dynamics in characteristic $2$ and point to concrete avenues for tightening bounds and formulating broader conjectures.

Abstract

We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.

A variant of Collatz's Conjecture over Binary Polynomials

TL;DR

This work studies a binary polynomial analogue of the Collatz Syracuse process over by defining parity via and constructing an associated dynamical system on polynomials, yielding sequences of even and odd polynomials. It proves a finite-length result (Theorem collatz0) with an exponential upper bound , based on degree convergence arguments and a tail linear system with a circulant matrix whose determinant is nonzero. Complementary computational data for degrees up to suggest the actual sequence lengths may grow polynomially in the degree and motivate conjectures such as and , along with patterns in special families like where . The results advance understanding of polynomial dynamics in characteristic and point to concrete avenues for tightening bounds and formulating broader conjectures.

Abstract

We study a natural analogue of Collatz's Conjecture for polynomials over .

Paper Structure

This paper contains 7 sections, 10 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['collatz0']}
  3. Some computer computations
  4. Example 1
  5. Other small lengths
  6. More small lengths
  7. Some technical remarks

Key Result

Theorem 1.2

Let $A$ be a nonzero binary polynomial. Then the sequences of polynomials obtained from Collatz transformations are of finite length $r_A$. More precisely, these sequences are where $m \in \mathbb{N}^*$ and $r_A = m+1 \leq 2^{\deg(A)-1}$.

Theorems & Definitions (20)

  • Conjecture 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • ...and 10 more