Uniform denominators in Hilbert's seventeenth problem
Uniform denominators in Hilbert's seventeenth problem
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DOI：
10.1007/bf02572604
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发表时间：
1995-12
影响因子：
0.8
通讯作者：
B. Reznick
中科院分区：
文献类型：
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作者：
B. Reznick
1888年，希尔伯特[H4]证明，如果n> 3，则存在实正半定形式p= p (xl,..., xn)，不能写成形式的平方和。他还证明，对于 n= 3，这种形式始终是有理函数的平方和。希尔伯特第十七个问题询问任意数量变量中的正半定形式是否一定是有理函数的平方和。 20世纪20年代，Artin利用Artin-Schreier实域理论，肯定地解决了希尔伯特第十七问题。这个证明没有建设性。几年后，Prlya [P1]在一个特殊情况下提出了具体的证明：如果p既是正定的又是偶数，那么对于足够大的r，p.(~-~xi2) r有正系数，每海单项式平方和也是如此。这一事实意味着 p 是具有公分母 (y'~ x2) r/2 的有理函数的平方和。 1940 年，Habicht [HI] 使用 Prlya 的结果将任意正定形式 p 写为两个单项式平方和的商。从他的证明可以看出，p 是具有正定分母的有理函数的平方和。在每种情况下，如果 p 具有有理系数，则单项式也如此。（参见 [H2，第 57-59,300-304 页]。）这些结果在指定分母的性质方面比 Artin 的结果更强，但较弱，因为它们仅适用于实正定形式。
In 1888, Hilbert [H4] proved that if n> 3, then there exist real positive semidefinite forms p= p (xl,..., xn) which cannot be written as a sum of squares of forms. He also proved that for n= 3 such a form is always a sum of squares of rational functions. Hilbert's Seventeenth Problem asked whether a positive semidefinite form in any number of variables must be a sum of squares of rational functions. In the 1920s, Artin solved Hilbert's Seventeenth Problem in the affirmative by using the Artin-Schreier theory of real fields. This proof was not constructive.A few years later, Prlya [P1] presented a concrete proof in one special case: if p is both positive definite and even, then for sufficiently large r, p.(~-~ xi2) r has positive coefficients, and so is per sea sum of squares of monomials. This fact implies that p is a sum of squares of rational functions with common denominator (y'~ x2) r/2. In 1940, Habicht [HI] used Prlya's result to write an arbitrary positive definite form p as a quotient of two sums of squares of monomials. It follows from his proof that p is a sum of squares of rational functions with positive definite denominator. In each case, if p has rational coefficients, then so do the monomials.(See [H2, pp. 57-59,300-304].) These results are stronger than Artin's in specifying the nature of the denominators, but weaker in that they only apply to real positive definite forms.