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Tropical varieties for exponential sums

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Abstract

We study the complexity of approximating complex zero sets of certain n-variate 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 \(\mathbb {R}^n\). In particular, we give an explicit upper bound on the Hausdorff distance: \(\Delta (R,T) =O\left( t^{3.5}/\delta \right) \), where t and \(\delta \) 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 \(\mathbb {R}\), whereas membership and distance queries for our polyhedral approximation T can be decided in polynomial-time for any fixed n.

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Notes

  1. [44] provides an excellent survey on undecidability, in the classical Turing model, geared toward non-experts in complexity theory.

  2. A preliminary version of Theorem 1.10 appeared in our December 2014 Math ArXiV preprint 1412.4423 and was presented by the first author at MEGA 2015 (June 16, University of Trento).

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Acknowledgements

We thank Timo de Wolff for pointing out Silipo’s work [51]. We also thank Pascal Koiran, Gregorio Malajovich, Jiří Matoušek, and Klaus Meer for useful discussions. Special thanks go to our referee for valuable commentary that significantly improved our paper.

In closing, we would like to remember our friend and colleague Joel Zinn: He was admired at Texas A&M (and far beyond) for his wisdom, warmth, humor, and kindness. He is sorely missed.

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Authors and Affiliations

  1. Technische Universität Berlin, Institut für Mathematik, Sekretariat MA 3-2, Strasse des 17. Juni 136 10623, Berlin, Germany

    Alperen A. Ergür

  2. Department of Mathematics, Texas A&M University, TAMU 3368, College Station, TX, 77843-3368, USA

    Grigoris Paouris & J. Maurice Rojas

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  1. Alperen A. Ergür
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  2. Grigoris Paouris
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Correspondence to J. Maurice Rojas.

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Communicated by Ngaiming Mok.

In memory of Joel Zinn (March 16, 1946–December 5, 2018), beloved friend and brilliant colleague.

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A.E. was partially supported by NSF Grant CCF-1409020, and Einstein Foundation, Berlin. J.M.R. was partially supported by NSF Grant CCF-1409020, and LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by ANR. G.P. was partially supported by BSF Grant 2010288 and NSF CAREER Grant DMS-1151711.

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Ergür, A.A., Paouris, G. & Rojas, J.M. Tropical varieties for exponential sums. Math. Ann. 377, 863–882 (2020). https://doi.org/10.1007/s00208-019-01808-5

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