Skip to main content
SpringerLink
Log in

Heights of Toric Varieties, Entropy and Integration over Polytopes

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

Included in the following conference series:

Abstract

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.

Burgos was partially supported by the MINECO research project MTM2013-42135-P. Philippon was partially supported by the CNRS project PICS “Géométrie diophantienne et calcul formel” and the ANR research project “Hauteurs, modularité, transcendance”. Sombra was partially supported by the MINECO research project MTM2012-38122-C03-02.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baldoni, V., Berline, N., De Loera, J., Köppe, M., Vergne, M.: How to integrate a polynomial over a simplex. Math. Comput. 80, 297–325 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bost, J.-B., Gillet, H., Soulé, C.: Heights of projective varieties and positive Green forms. J. Amer. Math. Soc. 7, 903–1027 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brion, M.: Points entiers dans les polyèdres convexes. Ann. Sci. École Norm. Sup. 21(4), 653–663 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Burgos-Gil, J.I., Philippon, P., Sombra, M.: Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights. Astérisque 360, Soc. Math., France (2014)

    MATH  Google Scholar 

  5. Cox, D.A., Little, J.D., Schenck, H.K.: Toric varieties, Grad. Stud. Math. 124, Amer. Math. Soc. (2011)

    Google Scholar 

  6. Fulton, W.: Introduction to Toric Varieties. Ann. of Math. Stud., vol. 131. Princeton University Press, Princeton (1993)

    Google Scholar 

  7. Guillemin, V.: Moment maps and combinatorial invariants of Hamiltonian \(\mathbb{T}^n\)-spaces. Progr. Math., vol. 122. Birkhäuser, Boston (1995)

    Google Scholar 

  8. Maillot, V.: Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables. Mém. Soc. Math. France 80 (2000)

    Google Scholar 

  9. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  10. Zhang, S.-W.: Small points and adelic metrics. J. Algebraic Geom. 4, 281–300 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3), Calle Nicolás Cabrera 15, Campus UAM, Cantoblanco, 28049, Madrid, Spain

    José Ignacio Burgos Gil

  2. Institut de Mathématiques de Jussieu – U.M.R. 7586 du CNRS, Équipe de Théorie des Nombres, BP 247, 4 Place Jussieu, 75005, Paris, France

    Patrice Philippon

  3. ICREA & Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Martín Sombra

Authors
  1. José Ignacio Burgos Gil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrice Philippon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martín Sombra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrice Philippon .

Editor information

Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Burgos Gil, J.I., Philippon, P., Sombra, M. (2015). Heights of Toric Varieties, Entropy and Integration over Polytopes. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25040-3_32

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25039-7

  • Online ISBN: 978-3-319-25040-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation