On irreducibility of prefixed algebraic sets in moduli spaces of prime degree polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917861753094144 |
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| author | Patra, Niladri |
| author_facet | Patra, Niladri |
| contents | Consider the moduli space, $\mathcal{M}_{d}$, of degree $d \geq 2$ polynomials over $\mathbb{C}$, with a marked critical point. Given $k \geq 0,\; p$ an odd prime, we show that the set $Σ_{k,1,p}$ of conjugacy classes of degree $p$ polynomials, for which the marked critical point is strictly $(k,1)$-preperiodic, is an irreducible quasi-affine algebraic set. Irreducibility of these sets was conjectured by Milnor, and has been proved for $p=3$ by Buff, Epstein and Koch.
We prove that the subspaces of $Σ_{k,1,p}$, that arise by varying the ramification index of the marked critical point all the way up to the unicritical case, are all irreducible subvarieties. Finally, using the irreducibility of $Σ_{k,1,p}$ we give a new and short proof of the fact that the set of all unicritical points of $Σ_{k,1,p}$ form one Galois orbit under the action of absolute Galois group of $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04778 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On irreducibility of prefixed algebraic sets in moduli spaces of prime degree polynomials Patra, Niladri Dynamical Systems 11R09 (Primary) 37F12, 37P45 (Secondary) Consider the moduli space, $\mathcal{M}_{d}$, of degree $d \geq 2$ polynomials over $\mathbb{C}$, with a marked critical point. Given $k \geq 0,\; p$ an odd prime, we show that the set $Σ_{k,1,p}$ of conjugacy classes of degree $p$ polynomials, for which the marked critical point is strictly $(k,1)$-preperiodic, is an irreducible quasi-affine algebraic set. Irreducibility of these sets was conjectured by Milnor, and has been proved for $p=3$ by Buff, Epstein and Koch. We prove that the subspaces of $Σ_{k,1,p}$, that arise by varying the ramification index of the marked critical point all the way up to the unicritical case, are all irreducible subvarieties. Finally, using the irreducibility of $Σ_{k,1,p}$ we give a new and short proof of the fact that the set of all unicritical points of $Σ_{k,1,p}$ form one Galois orbit under the action of absolute Galois group of $\mathbb{Q}$. |
| title | On irreducibility of prefixed algebraic sets in moduli spaces of prime degree polynomials |
| topic | Dynamical Systems 11R09 (Primary) 37F12, 37P45 (Secondary) |
| url | https://arxiv.org/abs/2305.04778 |