一类半正定多项式的配平方和算法  

AN ALGORITHM FOR SUMS OF SQUARES OF A CLASS OF POSITIVE SEMI-DEFINITE POLYNOMIALS

作　　者：李轶[1] 

机构地区：[1]中国科学院成都计算机应用研究所,成都610041

出　　处：《系统科学与数学》2008年第4期490-504,共15页Journal of Systems Science and Mathematical Sciences

基　　金：中国科学院知识创新重要方向性项目(KJCX2-YW-S02);国家重点基础研究发展规划基金(2004CB318003)资助课题.

摘　　要：以牛顿多胞型技术为基础,根据牛顿多胞型中的点与点之间的相关性,给出了直接搜索多项式配平方和所需的最基本的项集X_s的算法,利用精确的符号算法PCAD,可将一类半正定多项式配成平方和,并编写了Maple程序"ASSOS",实现了多项式配平方和的自动生成.由多项式结构的稀疏性,此算法更能有效处理稀疏多项式.这一算法提高了多项式配平方和的效率,从而促进了一类代数不等式可读性证明的自动生成.除此之外,还给出了多项式不能表示为平方和的一个充分条件.Based on Newton polytope technology, by means of the relation of points in Newton polytope, an algorithm is presented for representing a class of positive semi-definite polynomials as sums of squares. It＇s is very effective for a class of sparse polynomials. This algorithm improves the efficiency of sums of squares of real polynomial and advances the Automated Production of readable proof of a class of algebraic inequalities. In addition, a sufficient condition is derived to determine that a given polynomial cannot be represented as a sum of squares.

关 键 词：平方和 Gram矩阵方法 牛顿多胞型 稀疏多项式 Hilbertl7问题. 

分 类 号：O174.14[理学—数学] TP301.6[理学—基础数学]