Tropical Newton–Puiseux Polynomials

  • Dima GrigorievEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)


We introduce tropical Newton–Puiseux polynomials admitting rational exponents. A resolution of a tropical hypersurface is defined by means of a tropical Newton–Puiseux polynomial. A polynomial complexity algorithm for resolubility of a tropical curve is designed. The complexity of resolubility of tropical prevarieties of arbitrary codimensions is studied.


Tropical Newton–Puiseux polynomials Resolution of tropical hypersurfaces 



The author is grateful to the grant RSF 16-11-10075 and to MCCME for inspiring atmosphere.


  1. 1.
    Bittner, L.: Some representation theorems for functions and sets and their application to nonlinear programming. Numer. Math. 16, 32–51 (1970)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dobkin, D., Guibas, L., Hershberger, J., Snoeyink, J.: An efficient algorithm for finding the CSG representation of a simple polygon. Algorithmica 10, 1–23 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fukuda, K., Saito, S., Tamura, A.: Combinatorial face enumeration in arrangements and oriented matroids. Discret. Appl. Math. 31, 141–149 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Grigoriev, D., Podolskii, V.: Complexity of tropical and min-plus linear prevarieties. Comput. Complex. 24, 31–64 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kripfganz, A., Schulze, R.: Piecewise affine functions as a difference of two convex functions. Optimization 18, 23–29 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. AMS, Providence (2015)zbMATHGoogle Scholar
  7. 7.
    Ovchinnikov, S.: Max-min representation of piecewise linear functions. Beitr. Algebra Geom. 43, 297–302 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Theobald, T.: On the frontiers of polynomial computations in tropical geometry. J. Symb. Comput. 41, 1360–1375 (2006)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.CNRS, Mathématiques, Université de LilleVilleneuve d’AscqFrance

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