Dimension Quasi-polynomials of Inversive Difference Field Extensions with Weighted Translations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)


We consider Hilbert-type functions associated with finitely generated inversive difference field extensions and systems of algebraic difference equations in the case when the translations are assigned positive integer weights. We prove that such functions are quasi-polynomials that can be represented as alternating sums of Ehrhart quasi-polynomials of rational conic polytopes. In particular, we generalize the author’s results on difference dimension polynomials and their invariants to the case of inversive difference fields with weighted basic automorphisms.


Inversive difference field Inversive difference polynomials Characteristic set Dimension quasi-polynomial 


  1. 1.
    Barvinok, A.I.: Computing the Ehrhart polynomial of a convex lattice polytope. Discrete Comput. Geom. 12, 35–48 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barvinok, A.I., Pommersheim, J.E.: An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics. Math. Sci. Res. Inst. Publ., vol. 38, pp. 91–147. Cambridge Univ. Press (1999)Google Scholar
  3. 3.
    Barvinok, A.I.: Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75(255), 1449–1466 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dönch, C.: Standard bases in finitely generated difference-skew-differential modules and their application to dimension polynomials. Ph.D. thesis. Johannes Kepler University Linz, Research Institute for Symbolic Computation (RISC) (2012)Google Scholar
  5. 5.
    Ehrhart, E.: Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)zbMATHGoogle Scholar
  7. 7.
    Kondrateva, M.V., Levin, A.B., Mikhalev, A.V., Pankratev, E.V.: Differential and Difference Dimension Polynomials. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  8. 8.
    Levin, A.B.: Type and dimension of inversive difference vector spaces and difference algebras. VINITI, Moscow, Russia, no. 1606–82, pp. 1–36 (1982)Google Scholar
  9. 9.
    Levin, A.: Difference Algebra. Springer, New York (2008). CrossRefzbMATHGoogle Scholar
  10. 10.
    Levin, A.: Dimension polynomials of intermediate fields of inversive difference field extensions. In: Kotsireas, I.S., Rump, S.M., Yap, C.K. (eds.) MACIS 2015. LNCS, vol. 9582, pp. 362–376. Springer, Cham (2016). CrossRefGoogle Scholar
  11. 11.
    Levin, A.B.: Dimension polynomials of difference local algebras. Adv. Appl. Math. 72, 166–174 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Levin, A.B.: Difference dimension quasi-polynomials. Adv. Appl. Math. 89, 1–17 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Levin, A.B., Mikhalev, A.V.: Type and dimension of finitely generated G-algebras. Contemp. Math. 184, 275–280 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shananin, N.A.: On the unique continuation of solutions of differential equations with weighted derivatives. Sb. Math. 191(3–4), 431–458 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shananin, N.A.: On the partial quasianalyticity of distribution solutions of weakly nonlinear differential equations with weights assigned to derivatives. Math. Notes 68(3–4), 519–527 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.The Catholic University of AmericaWashington, DCUSA

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