Two model theoretic proofs of Rückert’s Nullstellensatz

Weispfenning, Volker

doi:10.1090/S0002-9947-1975-0369060-0

Two model theoretic proofs of Rückert’s Nullstellensatz
HTML articles powered by AMS MathViewer

by Volker Weispfenning PDF

Trans. Amer. Math. Soc. 203 (1975), 331-342 Request permission

Abstract:

Rückert’s Nullstellensatz for germs of analytic functions and its analogue for germs of real analytic functions are proved by a combination of nonstandard analysis with a model theoretic transfer principle. It is also shown that Rückert’s Nullstellensatz is constructive essentially relative to the Weierstrass preparation theorem.

References

D. W. Dubois, A nullstellensatz for ordered fields, Ark. Mat. 8 (1969), 111–114. MR 272761, DOI 10.1007/BF02589551

Local reality

Jean-Jacques Risler, Un théorème des zéros en géométrie analytique réelle, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1488–A1490 (French). MR 294686
Abraham Robinson, Germs, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 138–149. MR 0236427
Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
Abraham Robinson, Introduction to model theory and to the metamathematics of algebra, North-Holland Publishing Co., Amsterdam, 1963. MR 0153570
A. Seidenberg, Some remarks on Hilbert’s Nullstellensatz, Arch. Math. (Basel) 7 (1956), 235–240. MR 81929, DOI 10.1007/BF01900296
Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley-Los Angeles, Calif., 1951. 2nd ed. MR 0044472, DOI 10.1525/9780520348097
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0