Complexity Estimates for Representations of Schmüdgen Type

✍ Scribed by Elizabeth Mauch

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 100 KB
• Volume: 18
• Category: Article
• ISSN: 0885-064X
• DOI: 10.1006/jcom.2001.0616

✦ Synopsis

Let F={f 1 , ..., f j } and let K be a closed basic set in R n given by the polynomial inequalities f 1 \ 0, ..., f j \ 0. Let S{F} be the semiring generated by the f k and the squares in R[x 1 , ..., x n ]. For example, if F={f 1 } then S{F}=s 1 +s 2 f 1 , where s 1 , s 2 are sums of squares of polynomials. Schmüdgen has shown that if K is compact then any polynomial strictly positive on K belongs to S{F}. This paper develops a result of Schmüdgen type for functions in one dimension merely nonnegative on K. For this, it is necessary to add additional hypotheses, such as the proximity of complex zeros, to compensate for the loss of strict positivity necessary for Schmüdgen's result.