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A Ronkin Type Function for Coamoebas

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Abstract

The Ronkin function plays a fundamental role in the theory of amoebas. We introduce an analogue of the Ronkin function in the setting of coamoebas. It turns out to be closely related to a certain toric arrangement known as the shell of the coamoeba and we use our Ronkin type function to obtain some properties of it.

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Notes

  1. In perhaps more standard terminology, \(\mathcal {A}_f\) and \(\mathcal {A}_f'\) are the amoeba and coamoeba respectively of the Laurent polynomial \(\tilde{f}(\zeta ):=\sum _{\alpha \in A} c_{\alpha }\zeta ^{\alpha }\) on the complex torus \((\mathbb {C}^*)^n\).

  2. In other words, \(\eta \) is in \(K_u\) if and only if the Legendre transform of u at \(\eta \) is finite.

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Acknowledgments

We would like to thank Bo Berndtsson for valuable discussions on the topic of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Petter Johansson

  2. Department of Mathematical Sciences, Division of Mathematics, University of Gothenburg and Chalmers University of Technology, 412 96, Göteborg, Sweden

    Håkan Samuelsson Kalm

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  1. Petter Johansson
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  2. Håkan Samuelsson Kalm
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Correspondence to Håkan Samuelsson Kalm.

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Håkan Samuelsson Kalm was partially supported by the Swedish Research Council.

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Johansson, P., Samuelsson Kalm, H. A Ronkin Type Function for Coamoebas. J Geom Anal 27, 643–670 (2017). https://doi.org/10.1007/s12220-016-9693-z

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