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Aspects quantitatifs de Nullstellensätze et de Positivstellensätze. Nombres de Pythagore. (Quantitative aspects of Nullstellensätze and Positivstellensätze. Pythagoras numbers). (French) Zbl 0668.14001
The starting point of this paper is a generalized version of a theorem of A. Pfister given by L. Mahé concerning an upper bound of the number of squares occuring in the representation as a sum of squares of -1 in a k- algebra A of finite type over a real closed field k. The author extends this result to the larger class of k-algebras $$T_ d$$ (not necessarily of finite type) such that the transcendence degree of them is $$\leq d$$ (with $$d\geq 1$$ a fixed integer).
As corollaries one gets some quantitative forms of Nullstellensätze and Positivstellensätze. Finally, working with some modified Pythagoras numbers $$P^+(A)$$ and $$P^*(A)$$ of a commutative ring A, the author finds upper bounds for $$P^+(A)$$ and $$P^*(A)$$ when $$A\in T_ d$$. These kind of results are in contrast with the usual Pythagoras number P(A) which is known that it cannot be bounded with respect to the Krull dimension of A.
Reviewer: L.Bădescu

##### MSC:
 14A05 Relevant commutative algebra 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 14Pxx Real algebraic and real-analytic geometry 11E04 Quadratic forms over general fields
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