Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

Theresa C. Anderson, Evan M. O'Dorney

Abstract

We study the Galois group $G_f$ of a random polynomial $f$ of height at most $H$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$, which arise in a wide variety of applications. Our main result is a theorem of van der Waerden-Bhargava type: the probability that $G_f$ is not the full hyperoctahedral group $S_2 \wr S_n$ is $Θ(H^{-1}\log H)$, independent of $b$, with the leading-order group $G_1$ being of index $2$. This paper is a companion to a recent paper by the authors and Bertelli addressing reciprocal polynomials (i.e. the case $b = 1$).

Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

Abstract

We study the Galois group of a random polynomial of height at most in the family of polynomials of degree satisfying the twisted reciprocal relation , which arise in a wide variety of applications. Our main result is a theorem of van der Waerden-Bhargava type: the probability that is not the full hyperoctahedral group is , independent of , with the leading-order group being of index . This paper is a companion to a recent paper by the authors and Bertelli addressing reciprocal polynomials (i.e. the case ).
Paper Structure (12 sections, 11 theorems, 49 equations)

This paper contains 12 sections, 11 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. b-Reciprocal polynomials
  3. Notation
  4. Acknowledgments
  5. b-Reciprocal polynomials
  6. Counting G1-polynomials
  7. Counting G2-polynomials
  8. Counting G3-polynomials
  9. Other special polynomial families
  10. Reducible polynomials
  11. Imprimitive polynomials
  12. Primitive polynomials

Key Result

Theorem 1.1

Let $n \geq 1$ and $b \neq 0$ be integers. Let $\mathcal{E}_{n,b}(H)$ be the number of separable $b$-reciprocal polynomials $f$ of degree $2n$ with coefficients in $[-H,H]$ whose Galois group is not $S_2 \wr S_n$. Then, for fixed $n$ and $b$,

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2: ABORecipPolys, Theorem 3.4
  • Definition 2.3
  • Lemma 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 9 more