Papers › Limiting distributions of conjugate algebraic integers
Limiting distributions of conjugate algebraic integers
Bryce Joseph Orloski, Naser Talebizadeh Sardari
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Let Σ⊂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≥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→∞. We also prove our theorem for Σ⊂ℝ and open sets inside ℝ that recovers Smith's main theorem \cite{Smith} as special case. Given any finite field 𝔽_q and any integer n, our algorithm returns infinitely many abelian varieties over 𝔽_q which are not isogenous to the Jacobian of any curve over 𝔽_(qⁿ).
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