Efficient algorithms and applications of symbolic computation

符号计算的高效算法及应用

• 批准号: RGPIN-2015-06197
• 负责人: Zhang, Yang
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669736
• 关键词: Efficient; algorithms; applications; symbolic; computation

The overarching theme of this proposed program is the design, analysis and implementation of efficient algorithms for a number of important non-commutative algebraic structures in symbolic computation (also called computer algebra), such as Ore (skew) polynomials, polynomials over some non-commutative rings, generalized differential-difference algebras and matrices over these non-commutative algebras. Their study is motivated by many applications in coding theory, control theory, cryptography and engineering, etc.******The challenge in this area is that these non-commutative algebras generally have a much more complex structure when compared with commutative case.  In particular, many of the algorithmic breakthroughs in symbolic computation over the past three decades do not obviously apply in non-commutative domains. This is due both to the non-existence of particular mathematical properties in these rings or their inherently more difficult formulations. My research under this program will be to consider some of the most important computational problems in these non-commutative algebras at both a theoretical and practical level, which include fast algorithms for factoring Ore polynomials (both simple variable and multivariate cases), computing normal forms and inverses of matrices over Ore polynomial rings, and fast algorithms for computing Groebner bases in generalized differential-difference algebra and hope to achieve a unified and efficient approach to many well-known non-commutative domains. We also wish to give some applications, e.g.,  computing Gelfand-Kirillov dimensions,  and classifying Post-Lie algebras. The algorithmic advances of this proposal will be implemented in computer algebra software such as Maple, SAGE and Singular.  ****The proposed research plans all involve extensive training of highly qualified personnel (HQP) for their future positions in academia and industry.********

该计划的首要主题是设计，分析和实现一些重要的非交换代数结构的有效算法在符号计算（也称为计算机代数），如矿石（斜）多项式，多项式在一些非交换环，广义微分差分代数和矩阵在这些非交换代数。他们的研究动机是在编码理论，控制理论，密码学和工程等许多应用。这一领域的挑战在于，与交换代数相比，这些非交换代数通常具有更复杂的结构，特别是，过去三十年来在符号计算方面的许多算法突破显然不适用于非交换域。这是由于在这些环中不存在特定的数学性质或其固有的更困难的公式。我的研究在这个计划将考虑在这些非交换代数在理论和实践层面上的一些最重要的计算问题，其中包括快速算法的因式分解Ore多项式（单变量和多变量情况），计算Ore多项式环上矩阵的标准形和逆，和快速算法计算Groebner基的广义微分差分代数，并希望实现一个统一的和有效的方法，许多著名的非交换域。我们还希望提供一些应用程序，例如， 计算Gelfand-Kirillov维数，并分类后李代数。该提案的算法进步将在计算机代数软件中实现，如Maple，SAGE和Singular。* 拟议的研究计划都涉及对高素质人员（HQP）的广泛培训，以便他们将来在学术界和工业界任职。