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 -sets up to -equivariant -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 au-dessus de à -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.
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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
@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/} }
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%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] Torification and factorization of birational maps, J. Amer. Math. Soc., Volume 15 (2002) no. 3, p. 531-572 (electronic) | DOI
[2] 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] On periodic points, Ann. of Math. (2), Volume 81 (1965), pp. 82-99
[4] Arc-analytic functions, Invent. Math., Volume 101 (1990) no. 2, pp. 411-424 | DOI
[5] 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] 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] An inverse mapping theorem for blow-Nash maps on singular spaces (2014) (To appear in Nagoya Math. J.)
[8] On the degree of Igusa’s local zeta function, Amer. J. Math., Volume 109 (1987) no. 6, pp. 991-1008 | DOI
[9] Newton polyhedra and Igusa’s local zeta function, J. Number Theory, Volume 89 (2001) no. 1, pp. 31-64 | DOI
[10] Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Amer. Math. Soc., Volume 5 (1992) no. 4, pp. 705-720 | DOI
[11] Motivic Igusa zeta functions, J. Algebraic Geom., Volume 7 (1998) no. 3, pp. 505-537
[12] Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Volume 135 (1999) no. 1, pp. 201-232 | DOI
[13] Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J., Volume 99 (1999) no. 2, pp. 285-309 | DOI
[14] 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] Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology, Volume 41 (2002) no. 5, pp. 1031-1040 | DOI
[16] Motivic invariants of arc-symmetric sets and blow-Nash equivalence, Compos. Math., Volume 141 (2005) no. 3, pp. 655-688 | DOI
[17] Zeta functions and blow-Nash equivalence, Ann. Polon. Math., Volume 87 (2005), pp. 111-126 | DOI
[18] The corank and the index are blow-Nash invariants, Kodai Math. J., Volume 29 (2006) no. 1, pp. 31-40 | DOI
[19] 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] Motivic invariants of real polynomial functions and their Newton polyhedrons, Math. Proc. Cambridge Philos. Soc., Volume 160 (2016) no. 1, pp. 141-166
[21] Espaces d’arcs et invariants d’Alexander, Comment. Math. Helv., Volume 77 (2002) no. 4, pp. 783-820 | DOI
[22] 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] 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] 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] Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976) no. 1, pp. 1-31
[26] Une classification des singularités réelles, C. R. Acad. Sci. Paris Sér. A-B, Volume 288 (1979) no. 17, p. A809-A812
[27] On classification of real singularities, Invent. Math., Volume 82 (1985) no. 2, pp. 257-262 | DOI
[28] Ensembles semi-algébriques symétriques par arcs, Math. Ann., Volume 282 (1988) no. 3, pp. 445-462 | DOI
[29] Motivic measures, Astérisque (2002) no. 276, pp. 267-297 (Séminaire Bourbaki, Vol. 1999/2000)
[30] Virtual Betti numbers of real algebraic varieties, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 9, pp. 763-768 | DOI
[31] 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] Real algebraic manifolds, Ann. of Math. (2), Volume 56 (1952), pp. 405-421
[33] 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] Topology of injective endomorphisms of real algebraic sets, Math. Ann., Volume 328 (2004) no. 1-2, pp. 353-372 | DOI
[35] 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] Singularités à l’infini et intégration motivique, Bull. Soc. Math. France, Volume 140 (2012) no. 1, pp. 51-100
[37] 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] Zeta-function of monodromy and Newton’s diagram, Invent. Math., Volume 37 (1976) no. 3, pp. 253-262
[39] 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
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