Realization of decompositions of algebraic structures on real computer

该项目的目标包括两个目标：第一个目标是检验如何通过集中于代数结构的分解来在真实计算机上执行对代数结构的高/深数学运算。利用符号运算和代数运算，我们尝试实现与“分解”相关的数学运算。二是利用已实现的运算进行数学研究，作为计算工具。在计算机上实现的数学运算将支持数学家研究未解决的问题，并能产生一定的计算机辅助证明。扩展了这种计算的能力，我们将其应用于实际的工程问题。为了实现这些目标，我们选择了一些主题，为这些主题开发了有效/高效的算法，实现了这些算法，并在计算实验中测试了它们的能力。我们得到了令人满意的结果，为实现更高/更深的数学运算…找到了有前途的方法更多的颂词。(1)交换代数：对于有限域上多项式理想的素数分解，我们得到了一个有效的算法及其实际实现。对于含参数指数的多项式理想，我们基于“Groebner基的形式”定义了这类理想的稳定性，并在更简单的情况下给出了判定这种稳定性的完整算法。(2)高对称性交换代数：利用有理系数多项式的根的p-进近似和Galois群的信息，得到了计算有理系数多项式分裂域的一种实用方法。(3)非交换代数：我们得到了由格导出的顶点算子代数的不可约模分类的计算机辅助证明。(4)对数学和工程的支持：我们还得到了解决与代数曲线有关的未解猜想的计算机辅助证明，并成功地将Groebner基技术应用于控制理论中的问题。较少

The goal of the project consists of two aims : The first aim is to examine how high/deep mathematical operations on algebraic structures can be executed on real computer by concentrating on decompositions of algebraic structures. Using symbolic and algebraic computation, we try to realize the mathematical operations related to "decomposition". The second one is to utilize realized operations for studies on mathematics as computational tools. The realized mathematical operations on computer shall support mathematicians to investigate unsolved problems and it can produce certain computer-assisted-proofs. Extending the ability of such computations, we apply those to real engineering problems. For those aims, we selected a number of themes, for which we developed effective/efficient algorithms, implemented those, and examined their ability on computational experiments. We have obtained satisfactory results and found promising approaches for realization of more higher/deeper mathematical op … More erations. We list themes and give details for each :(1)Commutative Algebra :For prime decomposition of polynomial ideals over finite fields, we obtained an efficient algorithm and its practical implementation. For polynomial ideals with parametric exponents, we defined certain stability of those ideals based on "forms of Groebner bases", and gave a complete algorithm for deciding such stability for simpler cases. We applied "numeric-symbolic computation" to the CAD algorithm for quantifier elimination.(2)Commutative Algebra with High Symmetry :We obtained a practical method for computing the splitting field of a polynomial with rational coefficients by using p-adic approximations of its roots and information of its Galois group.(3)Non-Commutative Algebra :We obtained a computer-assisted-proof in classification of irreducible modules of the vertex operator algebra derived from a lattice.(4)Supports for Mathematics and Engineering :We also obtained a computer-assisted-proof in solving an unsolved conjecture related algebraic curves, and we also applied Groebner bases technique successfully to problems in control theory. Less