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Motivic integrals and functional equations
Author(s):
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
Posted:
May 9, 2008
MathSciNet review:
2381934
Retrieve article in:
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References |
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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:
-
- 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
 -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:
10.1090/S1061-0022-08-01010-8
PII:
S 1061-0022(08)01010-8
Keywords:
Motivic integration,
Milnor number,
motivic measure,
Grothendieck ring
Received by editor(s):
3/OCT/2006
Posted:
May 9, 2008
Additional Notes:
Supported by the grant NSh-4719.2006.1
Copyright of article:
Copyright
2008,
American Mathematical Society
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