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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Motivic integrals and functional equations


Author: E. Gorskii
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 4.
Journal: St. Petersburg Math. J. 19 (2008), 561-575
MSC (2000): Primary 32S45, 28B10
DOI: https://doi.org/10.1090/S1061-0022-08-01010-8
Published electronically: May 9, 2008
MathSciNet review: 2381934
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Abstract | References | Similar Articles | Additional Information

Abstract: A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived by using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant, and there is a simple recursive algorithm to find its coefficients. The method is fairly universal and gives, for example, equations for the integral corresponding to the intersection number over the space of pairs of arcs and over the space of unordered collections of arcs.


References [Enhancements On Off] (What's this?)

  • 1. J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201-232. MR 1664700 (99k:14002)
  • 2. S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, A power structure over the Grothendieck ring of varieties, Math. Res. Lett. 11 (2004), no. 1, 49-57. MR 2046199 (2004m:14038)
  • 3. M. Kapranov, The elliptic curve in the $ S$-duality theory and Eisenstein series for Kac-Moody groups, arXiv: math.AG/0001005.
  • 4. F. Heinloth, A note on functional equations for zeta functions with values in Chow motives, Ann. Inst. Fourier (Grenoble) 57 (2007), 1927-1945. arXiv: math.AG/0512237. MR 2377891
  • 5. V. I. Arnol'd, A. N. Varchenko, and S. M. Guseın-Zade, Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals, ``Nauka'', Moscow, 1984; English transl., Monogr. Math., vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 0755329 (86m:58026); MR 0966191 (89g:58024)

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Additional Information

E. Gorskii
Affiliation: Moscow State University and Independent University of Moscow, Russia
Email: gorsky@mccme.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01010-8
Keywords: Motivic integration, Milnor number, motivic measure, Grothendieck ring
Received by editor(s): October 3, 2006
Published electronically: May 9, 2008
Additional Notes: Supported by the grant NSh-4719.2006.1
Article copyright: © Copyright 2008 American Mathematical Society

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