Counting Polynomials via Galois Actions on Root Subsets

TL;DR

This paper develops new bounds on the number of integer polynomials with bounded height that have a Galois group acting on roots as a specific permutation group, advancing understanding of polynomial Galois groups.

Contribution

It introduces novel upper bounds for counting polynomials with prescribed Galois groups, covering various group actions including transitive, k-homogeneous, and k-transitive groups.

Findings

Derived bounds for transitive subgroups of wreath products.

Established bounds for k-homogeneous subgroups.

Analyzed groups in their regular permutation representations.

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 $(G,\Omega)$. 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.