Sadullaev, Azimbay On the Weierstrass preparation theorem. (English) Zbl 1439.32019 Ann. Pol. Math. 123, Part 2, 473-479 (2019). Summary: The well-known Weierstrass Preparation Theorem states that if \(f( z,w)\) is holomorphic at a point \(( z^0,w^0)\in \mathbb{C}_z^n\times{\mathbb{C}}_w\) and \(f( z^0,w^0)=0\), but \(f (z^0,w) \not \equiv 0\), then in some neighborhood \(U=V\times W\) of this point \(f\) is represented as \[f(z,w)= [{{ (w-{{w}^0})}^m}+{{c}_{m-1}} (z){{ (w-{{w}^0})}^{m-1}}+\dots +{{c}_0}(z)]\varphi (z,w),\] where \({c}_j (z) \in \mathcal{O}(V)\), \({c}_j (z^0)=0\) for \(j=0,1,\ldots ,m-1\) and \(\varphi \in \mathcal{O}(U), \varphi (z,w)\neq 0\). In this paper, a global multidimensional (in \(w)\) analogue of this theorem is proved without the condition \(f ({{z}^0},w)\not \equiv 0\). Cited in 1 Document MSC: 32A60 Zero sets of holomorphic functions of several complex variables 32U05 Plurisubharmonic functions and generalizations 32U15 General pluripotential theory Keywords:Weierstrass preparation theorem; pseudopolynomials; oscillatory integrals; analytic sets; pluripolar sets; Osgood’s counterexample PDFBibTeX XMLCite \textit{A. Sadullaev}, Ann. Pol. Math. 123, Part 2, 473--479 (2019; Zbl 1439.32019) Full Text: DOI References: [1] B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (in Russian). [2] I. A. Ikromov, Damped oscillatory integrals and maximal operators, Math. Notes 78 (2005), 773-790. · Zbl 1133.42033 [3] I. A. Ikromov and Sh. A. Muranov, Estimates of oscillatory integrals with a damping factor, Math. Notes 104 (2018), 218-230. · Zbl 1401.42015 [4] W. Osgood, Lehrbuch der Funktionentheorie, Bd. II, Teubner, Leipzig, 1929. · JFM 55.0171.02 [5] A. Sadullaev, Criteria of algebraicity of analytic sets, Funktsional. Anal. i Prilozhen. 6 (1972), no. 1, 85-86 (in Russian). · Zbl 0264.32002 [6] A. Sadullaev, Criteria of algebraicity of analytic sets, Kirensky Institute of Physics, Krasnoyarsk, 1976, 107-122 (in Russian). [7] C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in Rn, Invent. Math. 82 (1985), 543-556. · Zbl 0626.42009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.