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Ideals generated by exponential-polynomials. (English) Zbl 0586.32019
The authors introduce the concept of an ssd (strongly slowly decreasing) ideal in the ring of holomorphic functions on \({\mathbb{C}}^ n\) of exponential-polynomial type. This ring coincides with the space of Fourier transforms of measures of compact support on \({\mathbb{R}}^ n\). The reason for this is to study the sets on which such functions can be small. Problems of this type arise both in harmonic analysis and transcendental number theory. The spectrum of an ssd ideal is discrete; spectral synthesis (and even, under slightly more restrictive circumstances a Nullstellensatz) holds for these ideals. The objective of this paper is to find criteria to recognise whether a given ideal is ssd, to prove algebraic properties of these ideals and to apply these to the proof of geometric results related, mostly, to Schanuel’s conjecture. As an example we quote the following generalization of a theorem of Ritt: If F,G are exponential-polynomials on \({\mathbb{C}}^ n\), and if F/G is entire then there is a polynomial P factorizable into affine factors so that PF/G is also an exponential-polynomial.
Reviewer: S.Patterson

MSC:
32A38 Algebras of holomorphic functions of several complex variables
11J81 Transcendence (general theory)
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