Revista Matemática Iberoamericana

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Volume 30, Issue 4, 2014, pp. 1479–1487
DOI: 10.4171/RMI/822

Published online: 2014-12-15

A Nullstellensatz for Łojasiewicz ideals

Francesca Acquistapace[1], Fabrizio Broglia[2] and Andreea Nicoara[3]

(1) Università degli Studi di Pisa, Italy
(2) Università degli Studi di Pisa, Italy
(3) University of Pennsylvania, Philadelphia, USA

For an ideal of smooth functions $\mathfrak a$ that is either Łojasiewicz or weakly Łojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety $\mathcal I(\mathcal Z (\mathfrak a))$ in terms of the global Łojasiewicz radical and Whitney closure. We also prove that the Łojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to revisit Nullstellensatz results due to Bochnak and Adkins–Leahy and to resolve positively a modification of the Nullstellensatz conjecture due to Bochnak.

Keywords: Nullstellensatz, closed ideal, real Nullstellensatz, radical ideal, Łojasiewicz radical ideal, real ideal, real analytic ideal, Whitney closure, saturation of an ideal

Acquistapace Francesca, Broglia Fabrizio, Nicoara Andreea: A Nullstellensatz for Łojasiewicz ideals. Rev. Mat. Iberoam. 30 (2014), 1479-1487. doi: 10.4171/RMI/822