Limiting distributions of conjugate algebraic integers
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917631077908480 |
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| author | Orloski, Bryce Joseph Sardari, Naser Talebizadeh |
| author_facet | Orloski, Bryce Joseph Sardari, Naser Talebizadeh |
| contents | Let $Σ\subset \mathbb{C}$ be a compact subset of the complex plane, and $μ$ be a probability distribution on $Σ$. We give necessary and sufficient conditions for $μ$ to be the weak* limit of a sequence of uniform probability measures on a complete set of conjugate algebraic integers lying eventually in any open set containing $Σ$. Given $n\geq 0$, any probability measure $μ$ satisfying our necessary conditions, and any open set $D$ containing $Σ$, we develop and implement a polynomial time algorithm in $n$ that returns an integral monic irreducible polynomial of degree $n$ such that all of its roots are inside $D$ and their root distributions converge weakly to $μ$ as $n\to \infty$. We also prove our theorem for $Σ\subset \mathbb{R}$ and open sets inside $\mathbb{R}$ that recovers Smith's main theorem \cite{Smith} as special case. Given any finite field $\mathbb{F}_q$ and any integer $n$, our algorithm returns infinitely many abelian varieties over $\mathbb{F}_q$ which are not isogenous to the Jacobian of any curve over $\mathbb{F}_{q^n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02872 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Limiting distributions of conjugate algebraic integers Orloski, Bryce Joseph Sardari, Naser Talebizadeh Number Theory Let $Σ\subset \mathbb{C}$ be a compact subset of the complex plane, and $μ$ be a probability distribution on $Σ$. We give necessary and sufficient conditions for $μ$ to be the weak* limit of a sequence of uniform probability measures on a complete set of conjugate algebraic integers lying eventually in any open set containing $Σ$. Given $n\geq 0$, any probability measure $μ$ satisfying our necessary conditions, and any open set $D$ containing $Σ$, we develop and implement a polynomial time algorithm in $n$ that returns an integral monic irreducible polynomial of degree $n$ such that all of its roots are inside $D$ and their root distributions converge weakly to $μ$ as $n\to \infty$. We also prove our theorem for $Σ\subset \mathbb{R}$ and open sets inside $\mathbb{R}$ that recovers Smith's main theorem \cite{Smith} as special case. Given any finite field $\mathbb{F}_q$ and any integer $n$, our algorithm returns infinitely many abelian varieties over $\mathbb{F}_q$ which are not isogenous to the Jacobian of any curve over $\mathbb{F}_{q^n}$. |
| title | Limiting distributions of conjugate algebraic integers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2302.02872 |