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Sums of powers in rings and the real holomorphy ring. (English) Zbl 0922.12003
The real holomorphy ring of a commutative ring $$R$$ is defined to be the subring $$H(R)$$ of elements which are globally finite on the real spectrum of $$R$$. The authors present an extensive investigation of such real holomorphy rings. Suppose $$A$$ is a real affine domain over $${\mathbb R}$$, $$F$$ is the quotient field of $$A$$ and $$V$$ is the associated real affine variety. It is proved that $$A\subseteq H(F)$$ iff the set of all central real points of $$V$$ is compact. Moreover, if the subspace of singular, real points of V is compact then $$H(A)=A\cap H(F)$$. Considering the sequence $$H(R), H(H(R)), \ldots ,H^n(R)$$ of iterated real homomorphism rings, the authors show that if $$\dim R=d$$, then $$H^{d+1}(R)=H^d(R)$$.
For the semigroup of sums of higher powers holds the following: $\text{Sat}(1+\sum R^2)=\text{Sat}(1+\sum R^{2n})= \bigcap_{{\mathbf p}\in\text{Spec } R, {\mathbf p}\text{-real}} (R\setminus {\mathbf p}),$ where $$\text{Sat}(S)=\{s \in R \mid \exists_{s'\in S} ss'\in S\}$$ for any multiplicative semigroup $$S$$ in $$R$$.
The most complete results are obtained for rings satisfying $$1+\sum R^2 \subseteq R^*$$. In this case, every subring of $$R$$ containing $$H(R)$$ is a Prüfer ring in $$R$$. The connection between ideals and units of $$R$$ is established by the following isomorphism of groups: $(R^*\cap \sum R^{2n})/(R^*)^{2n}\cdot (H(R)^* \cap \sum R^2) \cong \text{Cl}(H(R),R),$ where $$\text{Cl}(H,R)$$ is the class group of invertible fractional ideals. The Waring $$n$$-constant is the smallest integer $$l$$ such that every element in $$R^*\cap \sum R^n$$ is a sum of $$l$$ $$n$$-th powers of elements of $$R$$, or $$\infty$$ if such $$l$$ does not exist. The authors prove that the Waring $$2n$$-constant for $$R$$ is finite iff the Waring 2-constant is finite. Moreover, for a certain class of rings an effective upper bound for the Waring $$2n$$-constant is determined.
Reviewer: M.Kula (Katowice)

##### MSC:
 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 11E81 Algebraic theory of quadratic forms; Witt groups and rings 14P99 Real algebraic and real-analytic geometry 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
##### Keywords:
real holomorphy ring; sum of powers; real spectrum
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