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arXiv · 2603.14617

Counting Polynomials via Galois Actions on Root Subsets

Or Ben-Porath
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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.

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Or Ben-Porath. 2026-03-15. Counting Polynomials via Galois Actions on Root Subsets. https://arxiv.org/abs/2603.14617

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