zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Atiyah M.–F., Introduction to commutative algebra (1969) · Zbl 0175.03601
[2] Bourbaki, N. 1961–1965. ”Algèbre Commutative”. Paris: Hermann. Ch. II
[3] Bourbaki N., Algèbre commutative (1983) · Zbl 0579.13001
[4] Bourbaki N., Algèbre (1965) · Zbl 0455.18010
[5] Bochnak J. Coste M. Roy M.F. Géométrie algébrique réelle Springer Verlag (à paraîltre)
[6] Choi M.D., J. Reine Angew. Math 336 pp 45– (1980)
[7] DOI: 10.1007/BFb0062251
[8] Godement R., Cours d’Algèebre (1966)
[9] Lam T.Y., The algebraic theory of quadratic forms (1973) · Zbl 0259.10019
[10] Lam T.Y., An introduction to real algebra (1984) · Zbl 0577.14016
[11] DOI: 10.2307/1969865 · Zbl 0052.03301
[12] DOI: 10.1007/BF01388792 · Zbl 0601.14019
[13] Matsumura H., Commutative algebra (1970)
[14] DOI: 10.1007/BF01425382 · Zbl 0222.10022
[15] Saliba C. Pour une classification des differentes formes de théorème des zéros et de théorèe des éléments positifs These 3ème cycle Université de Rennes 1983
[16] Simon O. Aspects quantitatifs de Stellensatze et algorithmes de multiplicativité des sommes de carrés Thèse 3ème cycle Université de Rennes 1987
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.