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Binary Cyclotomic Polynomials: Representation via Words and Algorithms

Antonio Cafure, Eda Cesaratto

Abstract

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector of coefficients of the polynomial as a word on a ternary alphabet $\{-1,0 ,+1\}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.

Binary Cyclotomic Polynomials: Representation via Words and Algorithms

Abstract

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector of coefficients of the polynomial as a word on a ternary alphabet . It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.

Paper Structure

This paper contains 5 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. A compact representation for binary cyclotomic polynomials
  4. Algorithms
  5. Conclusions and further work

Key Result

Theorem 2.1

Let $p < q$ be prime numbers. Consider the alphabet $\mathcal{A}=\{-1,0,1\}$ and the left circular permutation $\sigma$ defined in eq: def of circular permutation. With the $(p-1)$ words $\boldsymbol{d}_0, \boldsymbol{d}_1, \ldots \boldsymbol{d}_{p-2} \in \mathcal{ A}^p$, $\boldsymbol{d}_{0} = 1 Then, the following holds:

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof