Bochner transforms, perturbations and amoebae of holomorphic almost periodic mappings in tube domains

Abstract

We give an alternative representation of the closure of the Bochner transform of a holomorphic almost periodic mapping in a tube domain. For such mappings we introduce a new notion of amoeba and we show that, for mappings which are regular in the sense of Ronkin, this new notion agrees with Favorov's one. We prove that the amoeba complement of a regular holomorphic almost periodic mapping, defined on C n and taking its values in C m+1 , is a Henriques m-convex subset of R n . Finally, we compare some different notions of regularity.

The Ronkin number of an exponential sum

“…The following result, (a proof of which is available in Silipo [10] Proposition 3.2 and Corollary 3.9, or in Fabiano et al [1] Theorem 4.3 and Corollary 4.4) will be useful in the sequel.…”

“…Further details on such perturbations can be found in Silipo [10], or in Fabiano et al [1] for a generalization.…”

Higher convexity of coamoeba complements

“…Favorov [5] generalized amoebas to holomorphic almost periodic functions, and Henriques's result was extended to this setting in [4,17]. Rashkovskii [15] established a related result, the tube domain R n +iA c of the complement of an amoeba of a variety of codimension k is (n−k−1)-pseudoconvex, in the sense of Rothstein [16], which implies Mikhalkin's result on the absence of k-caps.…”

A Remark on Amoebas in Higher Codimensions

“…The notion of amoeba was adapted by Favorov [3] to zero sets of holomorphic almost periodic functions in a tube domain as "shadows" cast by the zero sets to the base of the domain; a precise definition is given in Section 4. In [2], Henriques' result was extended to amoebas of zero sets of so-called regular holomorphic almost periodic mappings. This was done by a reduction to the case considered in [10] where the proof was given by methods of algebraic geometry.…”