Solutions of systems of algebraic equations and linear maps on residue class rings

In this paper, we present new mathematical results and several new algorithm for solving a system of algebraic equations algebraically. We find that many ideal-theoretical arguments for the problem can be translated into their counterparts in the theory of linear maps. And by this translation, we succeed in giving a new description for the U-resultant and forms of solutions of systems straightforwardly. New algorithms proposed here apply algorithms of linear algebra to avoid repeated computations of Grobner bases under lexicographic order, and they require computation of a Grobner basis, under arbitrary order, only once in principle. The new algorithms improve the efficiency of computation.