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
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.