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On the Characteristic Polynomial of Linearized Polynomials (2506.16937v1)
Published 20 Jun 2025 in math.NT and cs.CC
Abstract: Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sumr_{i=0}a_iZ{qi}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large extension field $\mathbb F_{qn}\supseteq k$. Our algorithm has computational complexity of $O(n(\log(n))4)$ in terms of $\mathbb F_q$ operations with the implied constant depending only on $k$ and $r$. Up to logarithmic factors, and for linear maps represented by low degree polynomials, this provides a square root improvement over generic algorithms.