Amibes de sommes d’exponentielles

Silipo, James

doi:10.4153/cjm-2008-010-5

Cited by 5 publications

(17 citation statements)

References 9 publications

(22 reference statements)

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“…We thank Timo de Wolff for pointing out Silipo's work [Sil08]. We also thank Pascal Koiran, Gregorio Malajovich, Jiří Matoušek, and Klaus Meer for useful discussions.…”

Section: Acknowledgementsmentioning

confidence: 93%

Tropical varieties for exponential sums

2019

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We study the complexity of approximating complex zero sets of certain nvariate exponential sums. We show that the real part, R, of such a zero set can be approximated by the (n − 1)-dimensional skeleton, T , of a polyhedral subdivision of R n . In particular, we give an explicit upper bound on the Hausdorff distance: ∆(R, T ) = O t 3.5 /δ , where t and δ are respectively the number of terms and the minimal spacing of the frequencies of g. On the side of computational complexity, we show that even the n = 2 case of the membership problem for R is undecidable in the Blum-Shub-Smale model over R, whereas membership and distance queries for our polyhedral approximation T can be decided in polynomial-time for any fixed n.

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“…We thank Timo de Wolff for pointing out Silipo's work [Sil08]. We also thank Pascal Koiran, Gregorio Malajovich, Jiří Matoušek, and Klaus Meer for useful discussions.…”

Section: Acknowledgementsmentioning

confidence: 93%

Tropical varieties for exponential sums

2019

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“…The proof of (16) is straightforward, so let us show the equality (17). If F ∈ Exp(C n , C m ), the proof is formally the same as the one given in the Lemma 3.2 of the paper [6]. Let 1 ≤ ℓ ≤ m and let f ℓ be the ℓ-th component of F .…”

Section: Amoebaementioning

confidence: 93%

“…hold true, nevertheless they generally hold strictly (cf. [6]). Moreover Theorem 4.1 implies the equality…”

Section: Amoebaementioning

confidence: 99%

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

Fabiano¹,

Guenot²,

Silipo³

2007

Complex Variables and Elliptic Equations

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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.

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“…One needs then to bypass such a difficulty; see, for example, [12] for the construction of toric varieties associated to non rational fans. In this connection, we mention some references that might inspire ideas for deciding such conjectures about exponential sums [38,43,41,45,46,49,50,51,54,56]. Unfortunately, most of them do not really take into account the arithmetic constraints, and are more in the spirit of C. Moreno's papers [47,48].…”

Section: Some Other Miscellaneous Approachesmentioning

confidence: 99%

A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials

Yger

2012

From Fourier Analysis and Number Theory to Radon Transforms and Geometry

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Amibes de sommes d’exponentielles

Cited by 5 publications

References 9 publications

Tropical varieties for exponential sums

Tropical varieties for exponential sums

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

A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials

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