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A nonstandard proof of continuity of affine varieties

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Abstract

Extending the classical result that the roots of a polynomial with coefficients in \(\mathbf {C}\) are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if \(\mathcal {F}= \{f_{\lambda } :\lambda \in \Lambda \}\) is a set of polynomials in \(\mathbf {C}[t_1,\ldots , t_n]\) and if is a set of polynomials in such that \(g_{\lambda }\) is an infinitesimal deformation of \(f_{\lambda }\) for all \(\lambda \in \Lambda ,\) then the nonstandard affine variety is an infinitesimal deformation of the affine variety \(V(\mathcal {F}).\)

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Acknowledgements

This paper was inspired by several conversations with David Ross and by his recent paper [10]. I also thank Renling Jin for many discussions about nonstandard analysis over many years and for his paper [5].

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  1. Department of Mathematics, Lehman College (CUNY), Bronx, NY, 10468, USA

    Melvyn B. Nathanson

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  1. Melvyn B. Nathanson
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Correspondence to Melvyn B. Nathanson.

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Nathanson, M.B. A nonstandard proof of continuity of affine varieties. Arch. Math. 119, 583–592 (2022). https://doi.org/10.1007/s00013-022-01782-6

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  • DOI: https://doi.org/10.1007/s00013-022-01782-6

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