On the Generalization of Hilbert’s 17th Problem and Pythagorean Fields

The notion of preordering, which is a generalization of the notion of ordering, has been introduced by Serre. On the other hand, the notion of round quadratic forms has been introduced by Witt. Based on these ideas, it is here shown that 1) a field F is formally real n-pythagorean iff the nth radical, RnF is a preordering (Theorem 2), and 2) a field F is n-pythagorean iff for any n-fold Pfister form ρ. There exists an odd integer l(>1) such that l×ρ is a round quadratic form (Theorem 8). By considering upper bounds for the number of squares on Pfister’s interpretation, these results finally lead to the main result (Theorem 10) such that the generalization of pythagorean fields coincides with the generalization of Hilbert’s 17th Problem.

一类半正定多项式的平方和分解及其表达式的自动生成

建立了一个把半正定稀疏多项式表为多项式平方和的算法.这一算法依赖于Hilbert第17问题的一系列经典研究结果以及实闭域上量词消去的柱形代数剖分算法.该算法的机器实现为一类代数不等式可读性证明的自动生成提供了一种非常自然的途径.

曾炯与希尔伯特第17问题研究

通过对文献的考证和分析,对曾炯(1897—1940)与希尔伯特(D.Hilbert,1862—1943)第17问题的关系进行探究。结果表明,希尔伯特第17问题源于平方和问题,E.阿廷(E.Artin,1898—1962)在问题的定性方面给出了解答,但是问题的定量部分却是由费斯特(Pfister)彻底解决的,而在费斯特最初证明的最后关键一步应用了曾炯的结果。因此,曾炯对希尔伯特第17问题的定量解决做出了必要的贡献。