Abstract: The successive difference substitution method is studied. The author obtains an upper bound for successive difference substitution times of positive semi-definite polynomials on Rⁿ₊. A necessary and sufficient criterion for these polynomials is achieved by using this result. The procedure based on the above arguments always terminates. A new kind of successive difference substitution matrices are proposed.

Key words: successive difference substitution, successive difference substitution times, positive semi-definite, termination

摘要： 研究并发展逐次差分代换方法, 得到Rⁿ₊上正半定型差分代换次数的一个上界。 由此获得判定Rⁿ₊上 正半定型的充要条件。根据此充要条件建立的算法是必定能终止的。同时提出一类新的差分代换矩阵。

关键词: 逐次差分代换, 差分代换次数, 正半定型, 终止性