Hostname: page-component-f554764f5-wjqwx Total loading time: 0 Render date: 2025-04-13T12:14:59.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Motivic invariants of real polynomial functions and their Newton polyhedrons

Published online by Cambridge University Press:  26 November 2015

GOULWEN FICHOU  and
TOSHIZUMI FUKUI
GOULWEN FICHOU
Affiliation:
IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. e-mail: [email protected]
TOSHIZUMI FUKUI
Affiliation:
Department of Mathematics, Faculty of Science, Saitama University, Saitama, 338-8570, Japan. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give an expression of the motivic zeta function for a real polynomial function in terms of the Newton polyhedron of the function. As a consequence, we show that the weights are determined by the motivic zeta function for convenient weighted homogeneous polynomials in three variables. We apply this result to the blow-Nash equivalence.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 141 - 166
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Abderrahmane, O. M.Weighted homogeneous polynomials and blow-analytic equivalence, Singularity theory and its applications. Adv. Stud. Pure Math. 43 (Math. Soc. Japan, Tokyo, 2006), 333–345.CrossRefGoogle Scholar
[2]Arnold, V., Goussein–Zadé, S. and Varchenko, A.Singularity of Differentiable maps (Birkhauser, Boston 1985).CrossRefGoogle Scholar
[3]Denef, J. and Hoornaert, K.Newton polyhedra and Igusa's local zeta function. J. Number Theory 89 (2001), no. 1, 31–64.CrossRefGoogle Scholar
[4]Denef, J. and Loeser, F.Caractéristiques d'Euler–Poincaré, fonctions zêta locales et modifications analytiques. J. Amer. Math. Soc. 5 (1995), 705–720.Google Scholar
[5]Denef, J. and Loeser, F.Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135 (1999), no. 1, 201–232.CrossRefGoogle Scholar
[6]Denef, J. and Loeser, F.Geometry on arc spaces of algebraic varieties. Proceedings of 3rd European Congress of Mathematics, Barcelona. Progr. Math. 201 (Birkhauser, 2001), 327–348.CrossRefGoogle Scholar
[7]Fichou, G.Motivic invariants of arc-symmetric sets and Blow–Nash Equivalence. Compositio Math. 141 (2005), 655–688.CrossRefGoogle Scholar
[8]Fichou, G.Blow–Nash type of simple singularities. J. Math. Soc. Japan 60 no. 2 (2008), 445–470.CrossRefGoogle Scholar
[9]Fukui, T.Seeking invariants for blow-analytic equivalence. Compositio Math. 105 (1997), no. 1, 95–108.CrossRefGoogle Scholar
[10]Fukui, T., Koike, S. and Kuo, T.-C.Blow-analytic equisingularities, properties, problems and progress. Real Analytic and Algebraic Singularities (Fukuda, T., Fukui, T., Izumiya, S. and Koike, S., ed) Pitman Research Notes in Mathematics Series 381 (1998), pp. 8–29.Google Scholar
[11]Fukui, T. and Paunescu, L.On blow-analytic equivalence, Arc spaces and additive invariants in real algebraic geometry. Panoramas et Synthèses, SMF 26 (2008), 87–125.Google Scholar
[12]Guibert, G.Espaces d'arcs et invariants d'Alexander. Comment. Math. Helv. 77 (2002), no. 4, 783–820.CrossRefGoogle Scholar
[13]Koike, S. and Parusiński, A.Motivic-type invariants of blow-analytic equivalence. Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2061–2104.CrossRefGoogle Scholar
[14]Koike, S. and Parusiński, A.Blow-analytic equivalence of two variable real analytic function germs. J. Algebraic Geom. 19 (2010), no. 3, 439–472.CrossRefGoogle Scholar
[15]Kuo, T.-C.On classification of real singularities. Invent. Math. 82 (1985), 257–262.CrossRefGoogle Scholar
[16]McCrory, C. and Parusiński, A.Virtual Betti numbers of real algebraic varieties. C. R. Math. Acad. Sci. Paris 336 (2003), no. 9, 763–768.CrossRefGoogle Scholar
[17]Nishimura, T.Topological invariance of weights for weighted homogeneous singularities. Kodai Math. J. 9 (1986), no. 2, 188–190.CrossRefGoogle Scholar
[18]Saeki, O.Topological invariance of weights for weighted homogeneous isolated singularities in $\mathbb{C}$3. Proc. Ameri. Mathe. Soc. 103, No. 3 (July, 1988), 905–909.Google Scholar
[19]Saito, K.Quasihomogene isolierte Singularitaten von Hyperflachen. Invent. Math. 14 (1971), 123–142.CrossRefGoogle Scholar
[20]Yoshinaga, E. and Suzuki, M.Topological types of quasihomogeneous singularities in $\mathbb{C}$2. Topology 18 (1979), no. 2, 113–116.CrossRefGoogle Scholar