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Bézout identities for certain systems of exponential sums. (Identités de Bézout pour certains systèmes de sommes d’exponentielles.) (French) Zbl 0912.32003

The general problem of the Bézout identity is the following: For a given system of functions \((P_1,\dots,P_m)\) in \(\mathbb C^n \) (polynomials or entire functions) without common zeroes in \(\mathbb C^n \) find another system \((Q_1,\dots,Q_m)\) functions in the same class of functions such that \[ 1= Q_1P_1+\dots+Q_mP_m. \] The author solves this problem in the following special class of entire functions: Let \(\mathbb K\subset \mathbb C\) be a subfield and \(G\) be a subgroup of \(\mathbb K^n\) of finite type such that \(G\subset \mathbb R^n\) and \(\mathbb RG=\mathbb R^n\). Define the class \(\mathcal S_{\mathbb K,G}\) as the \(\mathbb K\)-algebra of all entire functions \(f\) which has the form of a finite exponential sum i.e. \[ f(z)=\sum_{\gamma\in G}a_\gamma e^{\langle\gamma,z\rangle},\qquad a_\gamma\in \mathbb K. \] Moreover, he proves a global Łojasiewicz inequality for a system of functions in the class \(\mathcal S_{\mathbb K,G}\).

MSC:

32A17 Special families of functions of several complex variables
32B05 Analytic algebras and generalizations, preparation theorems
32A15 Entire functions of several complex variables
32A27 Residues for several complex variables
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