快速分解稀疏多项式

IF 1.8 2区 数学 Q1 MATHEMATICS
Alexander Demin , Joris van der Hoeven
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引用次数: 0

摘要

考虑一个包含若干变量的稀疏多项式,其显式形式是有效域中带系数的非零项的和。在本文中,我们提出了分解这些多项式和相关任务的几种算法(如gcd计算,无平方分解,无内容分解和根提取)。我们的方法都是基于稀疏插值,但遵循两个主要的攻击路线:对变量数量的迭代和对单变量或双变量情况的更直接的简化。根据稀疏插值和求值的复杂度,给出了详细的概率复杂度界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Factoring sparse polynomials fast
Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms for factoring such polynomials and related tasks (such as gcd computation, square-free factorization, content-free factorization, and root extraction). Our methods are all based on sparse interpolation, but follow two main lines of attack: iteration on the number of variables and more direct reductions to the univariate or bivariate case. We present detailed probabilistic complexity bounds in terms of the complexity of sparse interpolation and evaluation.
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来源期刊
Journal of Complexity
Journal of Complexity 工程技术-计算机:理论方法
CiteScore
3.10
自引率
17.60%
发文量
57
审稿时长
>12 weeks
期刊介绍: The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.
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