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Tropical varieties for exponential sums

  • Alperen A. Ergür
  • Grigoris Paouris
  • J. Maurice RojasEmail author
Article

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.

Notes

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. 1.Technische Universität Berlin, Institut für MathematikBerlinGermany
  2. 2.Department of MathematicsTexas A&M University, TAMU 3368College StationUSA

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