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Construction of irreducible hypersurfaces with singular locus given in \(\mathbb{C}^n\). (Construction d’hypersurfaces irréductibles avec lieu singulier donné dans \(\mathbb{C}^n\).) (French) Zbl 0414.32004

Let \( S \) be an analytic set of codimension 22 in \( C^{n} \); we build irreducible hypersurfaces with singular locus \( S \), and with restricted growth. Using a counterexample to the transcendental Bézout problem, due to M. Cornalba and B. Shiffman, we find an irreducible curve of order \( O \) in \( \mathbb{C}^{2} \), and of infinite order singular locus. As an application, we also study some arithmetical properties of the convolution ring \( \delta^{\prime}\left(\mathbb{R}^{2}\right) \).

MSC:

32Sxx Complex singularities
14J17 Singularities of surfaces or higher-dimensional varieties
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
32C25 Analytic subsets and submanifolds
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References:

[1] [1] , , A counterexample to the “transcendental Bezout problem”, Ann. of Math. (2) 96 (1972), 402-406. · Zbl 0244.32006
[2] [2] , , Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. des Sc. Math., tome 102, fasc. n° 4 (1978), 305-330. · Zbl 0392.43013
[3] [3] , , , Irreducibility of certain entire functions with applications to harmonic analysis, Annals of Math., vol. 108, n° 3 (1978), 553-567. · Zbl 0402.32002
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