论文标题
计算确定性多项式时间中单变量的Igusa局部Zeta函数
Computing Igusa's local zeta function of univariates in deterministic polynomial-time
论文作者
论文摘要
Igusa的本地Zeta函数$ z_ {f，p}（s）$是计算积分根数的生成函数，$ n_ {k}（f）$，of $ f（\ mathbf x）\ bmod p^k $，用于所有$ k $。在分析数理论中，$ z_ {f，p} $是$ \ mathbb {q}（p^s）$中的一个有理函数。我们为单变量多项式$ f $提供了基本证明。我们的证明具有建设性，因为它为根数$ n_ {k}（f）$提供了封闭形式的表达式。 当我们的证明与最近的根计数算法结合使用（Dwivedi，Mittal，Saxena，CCC，2019年）时，会产生第一个确定性的poly（$ | f |，\ log p $）时间算法来计算$ z__ {f，p}（s）$。以前，只有在$ f $完全将$ \ mathbb {q} _p $完全拆分的情况下，才知道算法。它要求理性根本使用树生成树的概念（Zúñiga-Galindo，J.Int.Seq。，2003）。
Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression for the number of roots $N_{k}(f)$. Our proof, when combined with the recent root-counting algorithm of (Dwivedi, Mittal, Saxena, CCC, 2019), yields the first deterministic poly($|f|, \log p$) time algorithm to compute $Z_{f,p}(s)$. Previously, an algorithm was known only in the case when $f$ completely splits over $\mathbb{Q}_p$; it required the rational roots to use the concept of generating function of a tree (Zúñiga-Galindo, J.Int.Seq., 2003).
