半正定多项式的一个降次有理平方和表示算法  

An Algorithm to Represent Positive Semi-Definite Polynomials to Sum of Lader-Like Squares of Polynomials with Rational Coefficients

作　　者：黄勇[1,2] 曾振柄 杨路[4] 饶永生[1,2] HUANG Yong;ZENG Zhenbing;YANG Lu;RAO Yongsheng(Institute of Computing Science and Technology,Guangzhou University,Guangzhou 510006;Guangdong Province Research Center for Mathematical Educational Software,Guangzhou 510006;Department of Mathematics,Shanghai University,Shanghai 200444;Chengdu Institute of Computer Applications,Chinese Academy od Sciences,Chengdu 610041)

机构地区：[1]广州大学计算科技研究院,广州510006 [2]广东省数学教育软件工程技术研究中心,广州510006 [3]上海大学理学院数学系,上海200444 [4]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》2024年第5期1241-1271,共31页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(12171159)资助课题.

摘　　要：文章给出一个构造性算法,将一元半正定多项式表示为一些次数递降的多项式的平方和,当输入的多项式的系数是有理数时,该算法构造的降次多项式的系数也是有理数.文章还把这种方法推广到多元多项式情况,即如果该多项式有平方和表示,使用文章方法也能得到该半正定多元多项式的一个特殊的平方和分解.In this paper,the authors present a method to express a univariate positive semi-definite polynomial into the sum of lader-like squares,i.e.,squares of several polynomials which degrees are strictly decreasing.When the coefficients of the given polynomial are rational numbers,the coefficients of the degree-descending polynomials are also rational numbers.The authors have also extended the method to multi-variate polynomials.Namely,if a multi-variate polynomial has any sum of square representation,then a special representation with rational coefficients of this polynomial can be obtained using this method.

关 键 词：半正定多项式 平方和表示 降次多项式平方和 有理平方和 

分 类 号：O174.14[理学—数学]