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Counting Polynomials via Galois Actions on Root Subsets

Or Ben-Porath

Abstract

This paper studies the number of monic integer polynomials $f$ of height at most $H$ whose Galois group, endowed with the action on the roots, is isomorphic to a prescribed permutation group $(G,Ω)$. New upper bounds are obtained for several families of groups: transitive subgroups of the wreath product $S_m\wr S_r$ in the primitive action; $k$-homogeneous subgroups of $S_m$ in the action on $k$-subsets of $\{1,\ldots,m\}$; $k$-transitive subgroups of $S_m$ in the action on $k$-tuples of distinct elements of $\{1,\ldots,m\}$. Almost all finite groups in their regular permutation representation are also treated.

Counting Polynomials via Galois Actions on Root Subsets

Abstract

This paper studies the number of monic integer polynomials of height at most whose Galois group, endowed with the action on the roots, is isomorphic to a prescribed permutation group . New upper bounds are obtained for several families of groups: transitive subgroups of the wreath product in the primitive action; -homogeneous subgroups of in the action on -subsets of ; -transitive subgroups of in the action on -tuples of distinct elements of . Almost all finite groups in their regular permutation representation are also treated.
Paper Structure (4 sections, 18 theorems, 95 equations)

This paper contains 4 sections, 18 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Group actions
  3. Counting polynomials by group actions
  4. Proof of main results

Key Result

Theorem 1.1

Fix $\epsilon>0$, integers $m,r\ge 2$, and $1\le k\le m/2$. Let $G\le S_m\wr S_r$ be a primitive non-elemental permutation group of type $(r,m,k)$. Then

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 28 more