Limiting distributions of conjugate algebraic integers

arXiv2302.02872MaRDI QIDQ6508712

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Abstract: Let $SigmasubsetmathbbC$ be a compact subset of the complex plane, and $mu$ be a probability distribution on $Sigma$. We give necessary and sufficient conditions for $mu$ 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 $Sigma$. Given $ngeq0$, any probability measure $mu$ satisfying our necessary conditions, and any open set $D$ containing $Sigma$, 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 $mu$ as $noinfty$. We also prove our theorem for $SigmasubsetmathbbR$ and open sets inside $mathbbR$ 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 $mathbb{F}_{q^n}$.

Has companion code repository: https://github.com/Bryce-Orloski/Limiting-Distributions-of-Conjugate-Algebraic-Integers-Applications

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