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
黄勇 1曾振柄 2杨路 3饶永生1
作者信息
- 1. 广州大学计算科技研究院,广州 510006;广东省数学教育软件工程技术研究中心,广州 510006
- 2. 上海大学理学院数学系,上海 200444
- 3. 中国科学院成都计算机应用研究所,成都 610041
- 折叠
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.
Abstract
In this paper,the authors present a method to express a univariate posi-tive 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 polyno-mials 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Key words
Positive semi-definite polynomial/sum of squares of polynomials/sum of lader-like squares/rational SOS引用本文复制引用
出版年
2024
NETL
NSTL
万方数据