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原文摘要

For an $n\times n$ matrix over a field of characteristic two, let $Φ$ denote the sum of the odd permutation monomials. We give a deterministic algorithm that evaluates $Φ$ and all its first partial derivatives using $O(n^ω)$ field operations, where $2<ω\le3$ is any fixed admissible exponent for square matrix multiplication. As an application, we obtain faster randomized algorithms for shortest even directed cycles. Given an $n$-vertex directed graph whose shortest even cycle has length $\ell$, with high probability we compute $\ell$ and the union of the arcs of all shortest even cycles in $\widetilde O(n^ω\ell)$ time, and find one such cycle in $\widetilde O(n^ω\ell+n^2\ell^2)$ time. The resulting general bounds are $\widetilde O(n^{ω+1})$ and $\widetilde O(n^4)$, respectively, improving on the $\widetilde O(n^{ω+3})$ and $\widetilde O(n^{ω+4})$ bounds of Björklund, Husfeldt, and Kaski [BHK24].