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arXiv · 2409.08559

On the number of irreducible factors with a given multiplicity in function fields

Sourabhashis Das·Ertan Elma·Wentang Kuo·Yu-Ru Liu
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Abstract

Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $\omega_k(f)$ denote the number of distinct monic irreducible factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the first and the second moments of $\omega_k(f)$ with $k \geq 1$. Moreover, we prove that the function $\omega_1(f)$ has normal order $\log (\text{deg}(f))$ and also satisfies the Erd\H{o}s-Kac Theorem. Finally, we prove that the functions $\omega_k(f)$ with $k \geq 2$ do not have normal order.

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Sourabhashis Das, Ertan Elma, Wentang Kuo, Yu-Ru Liu. 2024-09-13. On the number of irreducible factors with a given multiplicity in function fields. https://doi.org/10.1016/j.ffa.2023.102281

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