| 摘要: |
| 在计算机辅助几何设计(CAGD)中,曲面曲线的凸性是一种重要的特性.旨在解决多项式的正性和凸性问题.凸性可以通过正性来解决.通过推广经典的Sturm定理,得到一种多项式正性的算法.由此提出了任意阶多项式为正的一个充要条件,也提出了一个实用的算法,从而可以只用此多项式的系数来表示得到的充要条件. |
| 关键词: 标准序列 最大公除数 正性 凸性 Bernstein-多项式 |
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| 基金项目:Supported by the Startup Scientific Research Fund for Returned Scholars from the Chinese Education Ministry (国家教育部留学回国人员科研启动基金); the Research Fund for Excellent Returned Scholars from the Chinese Academy of Sciences (中国科学院留学经费择优支持回国工作基金); the Post-Doctora |
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| On the Positivity and Convexity of Polynomials |
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ZHENG Jin-jin,CHEN Xiao-qun,ZHANG Jian-jun
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| Abstract: |
| The convexity of curves and surfaces is an important property in the field of Computer Aided Geometric Design (CAGD). This paper tries to tackle the positive and convex problem of polynomials. Convexity can be solved by positivity. An algorithm for the positivity of polynomials is developed by extending the classic Sturm theorem. Hence, a necessary and sufficient condition for the positivity of polynomials of arbitrary degree is presented in this paper. A practical algorithm to express this condition in terms of the coefficients of the polynomials is also given. |
| Key words: standard sequence greatest common divisor positivity convexity Bernstein-Polynomials |
