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Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 3, Pages 23–44 (Mi izv434)  

This article is cited in 16 scientific papers (total in 16 papers)

c-fans and Newton polyhedra of algebraic varieties

B. Ya. Kazarnovskii

Scientific Technical Centre "Informregistr"

Abstract: To every algebraic subvariety of a complex torus there corresponds a Euclidean geometric object called a c-fan. This correspondence determines an intersection theory for algebraic varieties. c-fans form a graded commutative algebra with visually defined operations. The c-fans of algebraic varieties lie in the subring of rational c-fans. It seems that other subrings may be used to construct an intersection theory for other categories of analytic varieties. We discover a relation between an old problem in the theory of convex bodies (the so-called Minkowski problem) and the ring of c-fans. This enables us to define a correspondence that sends any algebraic curve to a convex polyhedron in the space of characters of the torus.


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English version:
Izvestiya: Mathematics, 2003, 67:3, 439–460

Bibliographic databases:

UDC: 512.7+514.172
MSC: 52B20, 14M25, 14C17
Received: 15.06.2001

Citation: B. Ya. Kazarnovskii, “c-fans and Newton polyhedra of algebraic varieties”, Izv. RAN. Ser. Mat., 67:3 (2003), 23–44; Izv. Math., 67:3 (2003), 439–460

Citation in format AMSBIB
\by B.~Ya.~Kazarnovskii
\paper c-fans and Newton polyhedra of algebraic varieties
\jour Izv. RAN. Ser. Mat.
\yr 2003
\vol 67
\issue 3
\pages 23--44
\jour Izv. Math.
\yr 2003
\vol 67
\issue 3
\pages 439--460

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    This publication is cited in the following articles:
    1. B. Ya. Kazarnovskii, “Newton Polytopes, Increments, and Roots of Systems of Matrix Functions for Finite-Dimensional Representations”, Funct. Anal. Appl., 38:4 (2004), 256–266  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. B. Ya. Kazarnovskii, ““Newton polyhedra” of distributions”, Izv. Math., 68:2 (2004), 273–289  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. B. Ya. Kazarnovskii, “Multiplicative intersection theory and complex tropical varieties”, Izv. Math., 71:4 (2007), 673–720  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Proc. Steklov Inst. Math., 259 (2007), 16–34  mathnet  crossref  mathscinet  zmath  elib
    5. Esterov A., “Newton Polyhedra of Discriminants of Projections”, Discrete & Computational Geometry, 44:1 (2010), 96–148  crossref  mathscinet  zmath  isi  scopus
    6. A. Esterov, “Tropical varieties with polynomial weights and corner loci of piecewise polynomials”, Mosc. Math. J., 12:1 (2012), 55–76  mathnet  crossref  mathscinet
    7. Esterov A., “The Discriminant of a System of Equations”, Adv. Math., 245 (2013), 534–572  crossref  mathscinet  zmath  isi  elib  scopus
    8. Yang J.J., “Tropical Severi Varieties”, Port Math., 70:1 (2013), 59–91  crossref  mathscinet  zmath  isi  elib  scopus
    9. B. Ya. Kazarnovskii, “On the Action of the Complex Monge–Ampère Operator on Piecewise Linear Functions”, Funct. Anal. Appl., 48:1 (2014), 15–23  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. B. Ya. Kazarnovskii, “Action of the complex Monge–Ampère operator on piecewise-linear functions and exponential tropical varieties”, Izv. Math., 78:5 (2014), 902–921  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. B. Ya. Kazarnovskiǐ, A. G. Hovanskiǐ, “The tropical Noetherity and Gröbner bases”, St. Petersburg Math. J., 26:5 (2015), 797–811  mathnet  crossref  mathscinet  isi  elib  elib
    12. T. M. Sadykov, S. Tanabé, “Maximally reducible monodromy of bivariate hypergeometric systems”, Izv. Math., 80:1 (2016), 221–262  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Jensen A., Yu J., “Stable Intersections of Tropical Varieties”, J. Algebr. Comb., 43:1 (2016), 101–128  crossref  mathscinet  zmath  isi  scopus
    14. Yang J.J., “Secondary Fans and Tropical Severi Varieties”, Manuscr. Math., 149:1-2 (2016), 93–106  crossref  mathscinet  zmath  isi  scopus
    15. B. Ya. Kazarnovskii, “On the product of cocycles in a polyhedral complex”, Izv. Math., 81:2 (2017), 329–358  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. Esterov A., “Characteristic Classes of Affine Varieties and Plucker Formulas For Affine Morphisms”, J. Eur. Math. Soc., 20:1 (2018), 15–59  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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