Counting Polynomials via Galois Actions on Root Subsets
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915865109200896 |
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| author | Ben-Porath, Or |
| author_facet | Ben-Porath, Or |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14617 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Polynomials via Galois Actions on Root Subsets Ben-Porath, Or Number Theory 11R32, 11R09 (Primary) 11R45, 20B05 (Secondary) 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. |
| title | Counting Polynomials via Galois Actions on Root Subsets |
| topic | Number Theory 11R32, 11R09 (Primary) 11R45, 20B05 (Secondary) |
| url | https://arxiv.org/abs/2603.14617 |