Bulletin of the Belgian Mathematical Society - Simon Stevin

Nonstandard Generic Points

Guy Wallet

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Starting from the Zariski topology, a natural notion of nonstandard generic point is introduced in complex algebraic geometry. The existence of this kind of point is a strong form of the Nullstellensatz. This notion is connected with the classical concept of generic point in the spectrum $\text{Spec}({\cal A}_{n,\mathbb C})$ of the corresponding algebra ${\cal A}_{n,\mathbb C}$. The nonstandard affine space ${^*\mathbb C}^n$ appears as an affine unfolding of the geometric space $\text{Spec}({\cal A}_{n,\mathbb C})$. This affine space is the disjoint union of the sets whose elements are the nonstandard generic points of prime and proper ideals of ${\cal A}_{n,\mathbb C}$: this structure leads to the definition of algebraic points in ${^*\mathbb C}^n$. A natural extension to analytic points in ${^*\mathbb C}^n$ is given by Robinson's concept of generic point in local complex analytic geometry. The end of this paper is devoted to a generalization of this point of view to the real analytic case.

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Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 5 (2007), 1033-1057.

First available in Project Euclid: 1 February 2007

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Zentralblatt MATH identifier

Primary: 03H05: Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] 14P05: Real algebraic sets [See also 12D15, 13J30] 14P15: Real analytic and semianalytic sets [See also 32B20, 32C05] 14-99 32B05: Analytic algebras and generalizations, preparation theorems 32B10: Germs of analytic sets, local parametrization

Generic point nonstandard analysis Nullstellensatz Zariski topology


Wallet, Guy. Nonstandard Generic Points. Bull. Belg. Math. Soc. Simon Stevin 13 (2007), no. 5, 1033--1057. doi:10.36045/bbms/1170347824. https://projecteuclid.org/euclid.bbms/1170347824

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