Heights of Toric Varieties, Entropy and Integration over Polytopes

  • José Ignacio Burgos Gil
  • Patrice PhilipponEmail author
  • Martín Sombra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)


We present a dictionary between arithmetic geometry of toric varieties and convex analysis. This correspondence allows for effective computations of arithmetic invariants of these varieties. In particular, combined with a closed formula for the integration of a class of functions over polytopes, it gives a number of new values for the height (arithmetic analog of the degree) of toric varieties, with respect to interesting metrics arising from polytopes. In some cases these heights are interpreted as the average entropy of a family of random processes.


Line Bundle Toric Variety Closed Formula Average Entropy Toric Divisor 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • José Ignacio Burgos Gil
    • 1
  • Patrice Philippon
    • 2
    Email author
  • Martín Sombra
    • 3
  1. 1.Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3)Calle Nicolás Cabrera 15MadridSpain
  2. 2.Institut de Mathématiques de Jussieu – U.M.R. 7586 du CNRSÉquipe de Théorie des NombresParisFrance
  3. 3.ICREA & Departament d’Àlgebra i GeometriaUniversitat de BarcelonaBarcelonaSpain

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