Realization of decompositions of algebraic structures on real computer

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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