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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the volume of complex amoebas


Authors: Farid Madani and Mounir Nisse
Journal: Proc. Amer. Math. Soc. 141 (2013), 1113-1123
MSC (2010): Primary 14T05, 32A60
Published electronically: August 17, 2012
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Abstract: The paper deals with amoebas of $ k$-dimensional algebraic varieties in the complex algebraic torus of dimension $ n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of a $ k$-dimensional algebraic variety in $ (\mathbb{C}^*)^{n}$, with $ n\geq 2k$, is finite.


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Additional Information

Farid Madani
Affiliation: NWF I-Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: Farid.Madani@mathematik.uni

Mounir Nisse
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: nisse@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11394-2
PII: S 0002-9939(2012)11394-2
Keywords: Algebraic varieties, amoebas, logarithmic limit sets
Received by editor(s): February 11, 2011
Received by editor(s) in revised form: August 6, 2011
Published electronically: August 17, 2012
Additional Notes: The first author is supported by the Alexander von Humboldt Foundation.
The research of the second author is partially supported by NSF MCS grant DMS-0915245.
Dedicated: Dedicated to the memory of Mikael Passare (1959–2011)
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.