Dixon Resultant Research And New Algorithms

Solving the nonlinear algebraic equations is an important and fundamen-tal problem in constructive algebraic geometry. Its elimination methods includeWU method, Gr¨obner basis method and some resultant methods. Especially, theDixon resultant has been used widely in the resultant elimination to solve non-linear polynomial equations, such as in the high technology fields of AutomaticControl, Robotics, etc. So many researchers have studied its e?cient algorithmsin recent years.The theory research of Dixon resultant focuses on the three aspects: improv-ing e?ciency of computing Dixon resultant, finding extraneous factors of Dixonresultant and extracting a nontrivial minor from degenerate Dixon matrix, whichare leadingedge and hotspots. We have investigated the three problems deeplyand achieved some important breakthrough as follows:1. In order to improve e?ciency of computing Dixon resultant several meth-ods come into existence, such as Unknown-Order-Change method[8], Interpola-tion method[6], Fast-Matrix-Construction method[18, 31, 49, 57], Corner-Cuttingmethod[29], etc. Among these methods, Chionh's Fast-Matrix-Constructionmethod[31] has the lowest complexity and is the most e?cient, but which dealswith the case of three polynomial equations with two variables at most. We ex-tend the algorithm to the general case of n+1 polynomial equations in n variablesby n-degree Sylvester resultant, and based on which we propose the recursive al-gorithm of Dixon matrix, by which constructed the Dixon matrix of 9 Cyclicequations firstly.2. As for extraneous factors of Dixon resultant known as a persistent openproblem, though there are Corner-Cutting method[29], Support-Hull-Verticesmethod[32], Exposed-Points method[80] and so on, they could only handle somespecial systems. We introduce a new method for computing Dixon resultantby Dixon derived polynomials, by which the resulting Dixon resultant has noextraneous factors.3. There exist two methods to deal with degenerate Dixon matrix: KSYmethod[55] and Jacobian method[24]. KSY method was proposed in 1994 byYang Lu. Though it has been used widely, especially Nakos in U.S. NavalAcademy implemented it in Mathematica, there are no much progress in re-cent ten years. We propose a new method which is more understandable and canbe verified easily. The method shows that KSY condition is equivalent to thecondition that there exists a polynomial derived from Dixon matrix which is amonomial essentially.In total, firstly, we extend univariate Sylvester resultant to the general caseof n+1 polynomial equations in n variables. Secondly, we find out extraneousfactors of resultants. Thirdly, we make some improvements in degenerate matrixof Dixon resultant. What'more, we propose a series of fast algorithms of con-structing Dixon matrix, by which the Dixon matrices of some benchmark andengineering problems can be constructed. These methods and theories are veryimportant and helpful for solving the nonlinear algebraic equations by Dixonresultant.