Nonlinear Algebraic Polynomial System And Solving Geometry Constrained Problem

Solving nonlinear algebraic system is the kernel in nonlinear science. Many problems that come from engineering, mechanics or investigation produce some nonlinear algebraic systems, but it is a very difficult problem till now that one wants to solve those systems. Especially, geometry constrained problems that come from the engineering and mechanics usually produce nonlinear algebraic polynomial systems that have many equations or inequations with many variables and high degree at last. So it is too difficult to find their solutions without effective methods. In this paper, we have deeply investigated all these aspects as above. Under my supervisor professor Yang Lu's great help, we have got some main conclusions as follows: We have investigated the foundation theory in nonlinear algebraic system and given a foundation theorem that give an explicit expression about the basis in the dual linear space by the linear space of quotient ring of the polynomials ring. This theorem results in a very effective tool for solving nonlinear algebraic system; We have investigated the algorithm of solving the algebraic univariable polynomial equation with high degree. We got a real root located algorithm of polynomial equation in one unknown and high order; and also got a new constructive root-bound algorithm based on resultant for algebraic expression; Using the theory in distance geometry, we have given a systematic method and theory to produce a nonlinear algebraic system which have much fewer nonlinear algebraic polynomials than using Descartes' coordinate method directly, and a systematic procedure which may help us to covert the distances coordinate system to Descartes' coordinate of the geometry elements expediently. All of those results are very important to solving the geometry-constrained problem; Using constructive methods we have got some new results on the theory of embedded the geometry elements into Euclidean space; We have deeply discussed the numerical optimization theory, obtained some equivalent results; Combining the numerical algorithm with the symbolic algorithm, we have investigated the problem in solving nonlinear algebraic polynomial system, given some efficient algorithms for solving the system;We have designed the program with Maple. Using this program, we have solved three typical geometry constrained problems in computer, including Octahedron problem, Icositetrahedron problem and Packing problem. Indeed, we have given the complete symbolic results for some problems, which is impossible in the past.