Efficient algorithms and applications of symbolic computation

该计划的主要主题是设计、分析和实现符号计算（也称为计算机代数）中一些重要的非交换代数结构的有效算法，如偏多项式、一些非交换环上的多项式、广义微分-差分代数和这些非交换代数上的矩阵。他们的研究受到编码理论、控制理论、密码学和工程等领域的许多应用的推动******该领域的挑战在于，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，在过去的三十年中，符号计算中的许多算法突破并没有明显地应用于非交换域。这是由于在这些环中不存在特殊的数学性质，或者它们固有的更困难的公式。我在这个项目下的研究将是在理论和实践层面上考虑这些非交换代数中一些最重要的计算问题，包括分解矿石多项式（简单变量和多元情况）的快速算法，计算矿石多项式环上矩阵的范式和逆，以及广义微分-差分代数中计算Groebner基的快速算法，并希望对许多已知的非交换域实现统一和有效的方法。我们也希望给出一些应用，例如计算Gelfand-Kirillov维数和分类后李代数。本文的算法进步将在Maple、SAGE和Singular等计算机代数软件中实现。****所提出的研究计划都涉及对高素质人才（HQP）的广泛培训，为他们未来在学术界和工业界的职位********

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.********