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Non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with less than five prime factors

Olivier Rahavandrainy

TL;DR

This work completes the classification of non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with at most four irreducible factors by separating splitting and non-splitting cases. For splitting polynomials, it consolidates known results and reproduces the four admissible families, along with a complete list of perfect polynomials in this setting. For non-splitting polynomials, it shows $\omega(A)=2$ yields $A=P^hQ^k$ with $Q=P+1$ and $(h,k)=(2,2)$ or $(2^r-1,2^r-1)$, while $\omega(A)=3$ is impossible; for $\omega(A)=4$ it derives two structured families with $P\in\Omega_2$, $Q=P+1$, and $R,S$ among $\{P^3+P+1,P^3+P^2+1\}$, specifically with $(h,k,l,t)=(7,13,2,2),(13,7,2,2),(14,14,2,2)$. These results are supported by explicit Maple verification. The findings reveal both infinite families in the splitting case and a finite, highly structured set in the non-splitting case, advancing understanding of bi-unitary perfection over $\mathbb{F}_4$.

Abstract

We identify all non-splitting bi-unitary perfect polynomials over the field $\mathbb{F}_4$, which admit at most four irreducible divisors. There is an infinite number of such divisors.

Non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with less than five prime factors

TL;DR

This work completes the classification of non-splitting bi-unitary perfect polynomials over with at most four irreducible factors by separating splitting and non-splitting cases. For splitting polynomials, it consolidates known results and reproduces the four admissible families, along with a complete list of perfect polynomials in this setting. For non-splitting polynomials, it shows yields with and or , while is impossible; for it derives two structured families with , , and among , specifically with . These results are supported by explicit Maple verification. The findings reveal both infinite families in the splitting case and a finite, highly structured set in the non-splitting case, advancing understanding of bi-unitary perfection over .

Abstract

We identify all non-splitting bi-unitary perfect polynomials over the field , which admit at most four irreducible divisors. There is an infinite number of such divisors.

Paper Structure

This paper contains 11 sections, 25 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['casebupsplit']}
  4. Proof of Theorem \ref{['casebupnon-split']}
  5. Case $\omega(A)=2$
  6. Case $\omega(A)=3$
  7. Case $\omega(A)=4$
  8. Case $\deg(P) =1$
  9. Case $\deg(P) >1$
  10. End of the proof
  11. Maple Computations

Key Result

Theorem 1.1

Let $A = x^a(x+1)^b(x+\alpha)^c(x+\alpha+1)^d \in \mathbb{F}_4[x]$, where $a,b,c,d \in \mathbb{N}$ are not all odd. Then, $A$ is b.u.p if and only if one of the following conditions holds: i) $a=b=c=d=2$, ii) $a=b=2$ and $c=d=2^n-1$, for some $n \in \mathbb{N}$, iii) $a=b=2^n-1$ and $c=d=2$, for som

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 3.1
  • ...and 29 more