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Zbl 1087.32003
Acquistapace, Francesca; Broglia, Fabrizio; Fernando, José F.; Ruiz, Jesús M.
On the Pythagoras numbers of real analytic surfaces. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 38, No. 5, 751-772 (2005). ISSN 0012-9593

The following two theorems are the main results of the paper.\par Theorem 1.3. Let $X$ be a real analytic surface germ, and $f: X \to \Bbb R$ a positive semidefinite analytic function germ. Then there are analytic function germs $g, h_1, h_2, h_3, h_4 \in O (X)$ such that $$ g^2 f = h_1^2 + h_2^2 + h_3^2 + h_4^2$$ and $g$ is a sum of squares with $\{g = 0\} \subset \{f = 0\}.$\par Theorem 1.4. Let $X$ be a normal real analytic surface, and $f: X \to \Bbb R$ a positive semidefinite analytic function. Then there are analytic function germs $g, h_1, h_2, h_3, h_4, h_5 \in O (X)$ such that $$ g^2 f = h_1^2 + h_2^2 + h_3^2 + h_4^2 + h_5^2 $$ and $g$ is a sum of squares whose zero set $\{g = 0\}$ is a discrete subset of $\{f = 0\}.$ With the help of these theorems the authors have obtained the upper bound of the Pythagoras numbers of the ring of meromorphic function germs on a real analytic surface germ and the ring of meromorphic functions on a normal real analytic surface.
[Polina Z. Agranovich (Khar'kov)]

MSC 2000:: *32A20 Meromorphic functions (several variables)
32C07 Real-analytic sets
14P15 Real analytic and semianalytic sets

Keywords: Pythagoras number; analytic function; analytic function germ; meromorphic function; meromorphic function germ

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