On a motivic invariant of the arc-analytic equivalence
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 143-196.

To a Nash function germ, we associate a zeta function similar to the one introduced by J. Denef and F. Loeser. Our zeta function is a formal power series with coefficients in the Grothendieck ring of 𝒜𝒮-sets up to * -equivariant 𝒜𝒮-bijections over * , an analog of the Grothendieck ring constructed by G. Guibert, F. Loeser and M. Merle. This zeta function generalizes the previous construction of G. Fichou but thanks to its richer structure it allows us to get a convolution formula and a Thom–Sebastiani type formula.

We show that our zeta function is an invariant of the arc-analytic equivalence, a version of the blow-Nash equivalence of G. Fichou. The convolution formula allows us to obtain a partial classification of Brieskorn polynomials up to arc-analytic equivalence by showing that the exponents are arc-analytic invariants.

À un germe Nash, nous associons une fonction zêta similaire à la fonction zêta motivique de J. Denef et F. Loeser. Il s’agit d’une série formelle à coefficients dans un anneau de Grothendieck des ensembles 𝒜𝒮 au-dessus de * à 𝒜𝒮-bijection * -équivariante près. Cet anneau de Grothendieck est analogue à celui construit par G. Guibert, F. Loeser et M. Merle. Cette fonction zêta généralise les précédentes constructions de G. Fichou. Sa richesse algébrique permet d’obtenir une formule de convolution ainsi qu’une formule de type Thom–Sebastiani.

On démontre que la fonction zêta considérée dans cet article est un invariant de l’équivalence arc-analytique, une caractérisation de l’équivalence blow-Nash de G. Fichou. La formule de convolution permet d’obtenir une classification partielle des polynômes de Brieskorn à équivalence arc-analytique près. Plus précisément, on montre que le type arc-analytique d’un tel polynôme détermine ses exposants.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3078
Classification: 14P20, 14E18, 14B05
Keywords: real singularities, Nash functions, motivic integration, arc-analytic functions, blow-Nash equivalence, arc-analytic equivalence
Mot clés : singularités réelles, fonctions Nash, intégration motivique, fonctions analytiques par arcs, équivalence blow-Nash, équivalence arc-analytique

Campesato, Jean-Baptiste 1

1 Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351 06100 Nice (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2017__67_1_143_0,
     author = {Campesato, Jean-Baptiste},
     title = {On a motivic invariant of the arc-analytic equivalence},
     journal = {Annales de l'Institut Fourier},
     pages = {143--196},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     doi = {10.5802/aif.3078},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3078/}
}
TY  - JOUR
AU  - Campesato, Jean-Baptiste
TI  - On a motivic invariant of the arc-analytic equivalence
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 143
EP  - 196
VL  - 67
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3078/
DO  - 10.5802/aif.3078
LA  - en
ID  - AIF_2017__67_1_143_0
ER  - 
%0 Journal Article
%A Campesato, Jean-Baptiste
%T On a motivic invariant of the arc-analytic equivalence
%J Annales de l'Institut Fourier
%D 2017
%P 143-196
%V 67
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3078/
%R 10.5802/aif.3078
%G en
%F AIF_2017__67_1_143_0
Campesato, Jean-Baptiste. On a motivic invariant of the arc-analytic equivalence. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 143-196. doi : 10.5802/aif.3078. https://aif.centre-mersenne.org/articles/10.5802/aif.3078/

[1] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Włodarczyk, Jarosław Torification and factorization of birational maps, J. Amer. Math. Soc., Volume 15 (2002) no. 3, p. 531-572 (electronic) | DOI

[2] Arnold, Vladimir Igorevich; Gusein-Zade, Sabir M.; Varchenko, Alexander Nikolaevich Singularities of differentiable maps. Volume 2, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012, x+492 pages (Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous and revised by the authors and James Montaldi, Reprint of the 1988 translation)

[3] Artin, Michael; Mazur, Barry On periodic points, Ann. of Math. (2), Volume 81 (1965), pp. 82-99

[4] Bierstone, Edward; Milman, Pierre D. Arc-analytic functions, Invent. Math., Volume 101 (1990) no. 2, pp. 411-424 | DOI

[5] Bierstone, Edward; Milman, Pierre D. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., Volume 128 (1997) no. 2, pp. 207-302 | DOI

[6] Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36, Springer-Verlag, Berlin, 1998, x+430 pages (Translated from the 1987 French original, Revised by the authors) | DOI

[7] Campesato, Jean-Baptiste An inverse mapping theorem for blow-Nash maps on singular spaces (2014) (To appear in Nagoya Math. J.)

[8] Denef, Jan On the degree of Igusa’s local zeta function, Amer. J. Math., Volume 109 (1987) no. 6, pp. 991-1008 | DOI

[9] Denef, Jan; Hoornaert, Kathleen Newton polyhedra and Igusa’s local zeta function, J. Number Theory, Volume 89 (2001) no. 1, pp. 31-64 | DOI

[10] Denef, Jan; Loeser, François Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Amer. Math. Soc., Volume 5 (1992) no. 4, pp. 705-720 | DOI

[11] Denef, Jan; Loeser, François Motivic Igusa zeta functions, J. Algebraic Geom., Volume 7 (1998) no. 3, pp. 505-537

[12] Denef, Jan; Loeser, François Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Volume 135 (1999) no. 1, pp. 201-232 | DOI

[13] Denef, Jan; Loeser, François Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J., Volume 99 (1999) no. 2, pp. 285-309 | DOI

[14] Denef, Jan; Loeser, François Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000) (Progr. Math.), Volume 201, Birkhäuser, Basel, 2001, pp. 327-348

[15] Denef, Jan; Loeser, François Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology, Volume 41 (2002) no. 5, pp. 1031-1040 | DOI

[16] Fichou, Goulwen Motivic invariants of arc-symmetric sets and blow-Nash equivalence, Compos. Math., Volume 141 (2005) no. 3, pp. 655-688 | DOI

[17] Fichou, Goulwen Zeta functions and blow-Nash equivalence, Ann. Polon. Math., Volume 87 (2005), pp. 111-126 | DOI

[18] Fichou, Goulwen The corank and the index are blow-Nash invariants, Kodai Math. J., Volume 29 (2006) no. 1, pp. 31-40 | DOI

[19] Fichou, Goulwen Blow-Nash types of simple singularities, J. Math. Soc. Japan, Volume 60 (2008) no. 2, pp. 445-470 http://projecteuclid.org/euclid.jmsj/1212156658

[20] Fichou, Goulwen; Fukui, Toshizumi Motivic invariants of real polynomial functions and their Newton polyhedrons, Math. Proc. Cambridge Philos. Soc., Volume 160 (2016) no. 1, pp. 141-166

[21] Guibert, Gil Espaces d’arcs et invariants d’Alexander, Comment. Math. Helv., Volume 77 (2002) no. 4, pp. 783-820 | DOI

[22] Guibert, Gil; Loeser, François; Merle, Michel Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink, Duke Math. J., Volume 132 (2006) no. 3, pp. 409-457 | DOI

[23] Guillén, Francisco; Navarro Aznar, Vicente Un critère d’extension des foncteurs définis sur les schémas lisses, Publ. Math. Inst. Hautes Études Sci. (2002) no. 95, pp. 1-91 | DOI

[24] Koike, Satochi; Parusiński, Adam Motivic-type invariants of blow-analytic equivalence, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 7, pp. 2061-2104 http://aif.cedram.org/item?id=AIF_2003__53_7_2061_0

[25] Kouchnirenko, A. G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976) no. 1, pp. 1-31

[26] Kuo, Tzee Char Une classification des singularités réelles, C. R. Acad. Sci. Paris Sér. A-B, Volume 288 (1979) no. 17, p. A809-A812

[27] Kuo, Tzee Char On classification of real singularities, Invent. Math., Volume 82 (1985) no. 2, pp. 257-262 | DOI

[28] Kurdyka, Krzysztof Ensembles semi-algébriques symétriques par arcs, Math. Ann., Volume 282 (1988) no. 3, pp. 445-462 | DOI

[29] Looijenga, Eduard Motivic measures, Astérisque (2002) no. 276, pp. 267-297 (Séminaire Bourbaki, Vol. 1999/2000)

[30] McCrory, Clint; Parusiński, Adam Virtual Betti numbers of real algebraic varieties, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 9, pp. 763-768 | DOI

[31] McCrory, Clint; Parusiński, Adam The weight filtration for real algebraic varieties, Topology of stratified spaces (Math. Sci. Res. Inst. Publ.), Volume 58, Cambridge Univ. Press, Cambridge, 2011, pp. 121-160

[32] Nash, Jr., John F. Real algebraic manifolds, Ann. of Math. (2), Volume 56 (1952), pp. 405-421

[33] Nash, Jr., John F. Arc structure of singularities, Duke Math. J., Volume 81 (1995) no. 1, p. 31-38 (1996) (A celebration of John F. Nash, Jr.) | DOI

[34] Parusiński, Adam Topology of injective endomorphisms of real algebraic sets, Math. Ann., Volume 328 (2004) no. 1-2, pp. 353-372 | DOI

[35] Quarez, Ronan Espace des germes d’arcs réels et série de Poincaré d’un ensemble semi-algébrique, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 1, pp. 43-68 http://aif.cedram.org/item?id=AIF_2001__51_1_43_0

[36] Raibaut, Michel Singularités à l’infini et intégration motivique, Bull. Soc. Math. France, Volume 140 (2012) no. 1, pp. 51-100

[37] Steenrod, Norman The topology of fibre bundles, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999, viii+229 pages (Reprint of the 1957 edition, Princeton Paperbacks)

[38] Varchenko, Alexander Nikolaevich Zeta-function of monodromy and Newton’s diagram, Invent. Math., Volume 37 (1976) no. 3, pp. 253-262

[39] Whitney, Hassler Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 205-244

Cited by Sources: