A softly optimal Monte Carlo algorithm for solving bivariate polynomial systems over the integers

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• 作者：Esmaeil Mehrabi ^{emehrab@uwo.ca" class="auth_mail" title="E-mail the corresponding author} ; É ; ric Schost ; ^{eschost@uwo.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Bivariate system ; Complexity ; Algorithm
• 刊名：Journal of Complexity
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：34
• 期：Complete
• 页码：78-128
• 全文大小：696 K

文摘

We give an algorithm for the symbolic solution of polynomial systems in Z[X,Y]$\mathbb{Z}[X,Y]$. Following previous work with Lebreton, we use a combination of lifting and modular composition techniques, relying in particular on Kedlaya and Umans’ recent quasi-linear time modular composition algorithm.

The main contribution in this paper is an adaptation of a deflation algorithm of Lecerf, that allows us to treat singular solutions for essentially the same cost as the regular ones. Altogether, for an input system with degree d$d$ and coefficients of bit-size h$h$, we obtain Monte Carlo algorithms that achieve probability of success at least 1−1/2^P$1-1/2^{\mathcal{P}}$, with running time bit operations, for any ε>0$\epsilon >0$, where the notation indicates that we omit polylogarithmic factors.