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October 2021 A Nullstellensatz for ideals of $C^\infty$ functions in dimension 2
Hirofumi KONDO
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Hokkaido Math. J. 50(3): 455-462 (October 2021). DOI: 10.14492/hokmj/2019-201

Abstract

Suppose that an ideal $J$ of $C^\infty$ functions on an open subset of $\mathbf{R}^2$ is a Łojasiewicz ideal. We describe the set of $C^\infty$ functions vanishing on the zeros of $J$ explicitly using $J$ in an open neighborhood of each point in zeros of $J$, it can be obtained by taking real radical and closure starting from $J$ repeatedly for a finite number of times. This gives an another affirmative answer to Bochnak's conjecture in dimension 2, which is first done by Risler.

Acknowledgment

The author would like to express my sincere gratitude to Professor Shuzo Izumi for his helpful advice and encouragement. The author would like to thank the referees for their helpful suggestions.

Citation

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Hirofumi KONDO. "A Nullstellensatz for ideals of $C^\infty$ functions in dimension 2." Hokkaido Math. J. 50 (3) 455 - 462, October 2021. https://doi.org/10.14492/hokmj/2019-201

Information

Received: 18 November 2019; Revised: 18 March 2020; Published: October 2021
First available in Project Euclid: 17 December 2021

Digital Object Identifier: 10.14492/hokmj/2019-201

Subjects:
Primary: 26E05 , 26E10 , 46E25
Secondary: 11E25 , 14P15 , 32C05

Keywords: closed ideal , Łojasiewicz ideal , Nullstellensatz , real radical , zero property

Rights: Copyright c 2021 Hokkaido University, Department of Mathematics

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Vol.50 • No. 3 • October 2021
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