On the volume of complex amoebas
Authors:
Farid Madani and Mounir Nisse
Journal:
Proc. Amer. Math. Soc. 141 (2013), 11131123
MSC (2010):
Primary 14T05, 32A60
Published electronically:
August 17, 2012
MathSciNet review:
3008859
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Abstract 
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Additional Information
Abstract: The paper deals with amoebas of dimensional algebraic varieties in the complex algebraic torus of dimension . 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 dimensional algebraic variety in , with , is finite.
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 G. Mikhalkin, Decomposition into pairsofpants for complex algebraic hypersurfaces, Topology 43 (2004), 10351065. MR 2079993 (2005i:14055)
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 G. Mikhalkin, Enumerative tropical algebraic geometry in , J. Amer. Math. Soc. 18 (2005), 313377. MR 2137980 (2006b:14097)
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 G. Mikhalkin, Real algebraic curves, moment map and amoebas, Ann. of Math. (2) 151 (2000), 309326. MR 1745011 (2001c:14083)
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 G. Mikhalkin and H. Rullgård, Amoebas of maximal area, Int. Math. Res. Notices 9 (2001), 441451. MR 1829380 (2002b:14079)
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 M. Nisse and M. Passare, (Co)Amoebas of linear spaces, arXiv:1205.2808.
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 M. Nisse and F. Sottile, The phase limit set of a variety, Algebra and Number Theory, to appear.
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 M. Passare and H. Rullgård, Amoebas, MongeAmpère measures, and triangulations of the Newton polytope, Duke Math. J. 121 (2004), 481507. MR 2040284 (2005a:32005)
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 L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2004. MR 2205865 (2006i:92002)
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 J. RichterGebert, B. Sturmfels and T. Theobald, First steps in tropical geometry, Idempotent mathematics and mathematical physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005, 289317. MR 2149011 (2006d:14073)
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Additional Information
Farid Madani
Affiliation:
NWF IMathematik, Universität Regensburg, 93040 Regensburg, Germany
Email:
Farid.Madani@mathematik.uni
Mounir Nisse
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 778433368
Email:
nisse@math.tamu.edu
DOI:
http://dx.doi.org/10.1090/S000299392012113942
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 DMS0915245.
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.
