Skip to main content
SpringerLink
Log in

Solutions of quasianalytic equations

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

The article develops techniques for solving equations \(G(x,y)=0\), where \(G(x,y)=G(x_1,\ldots ,x_n,y)\) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy–Carleman class, or the class of \({\mathcal C}^\infty \) functions definable in a polynomially-bounded o-minimal structure). We show that, if \(G(x,y)=0\) has a formal power series solution \(y=H(x)\) at some point a, then H is the Taylor expansion at a of a quasianalytic solution \(y=h(x)\), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acquistapace, F., Broglia, F., Bronshtein, M., Nicoara, A., Zobin, N.: Failure of the Weierstrass preparation theorem in quasi-analytic Denjoy–Carleman rings. Adv. Math. 258, 397–413 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. I. Biborski, On the geometric and differential properties of closed sets definable in quasianalytic structures, preprint (2015). arXiv:1511.05071v1 [math.AG]

  3. Bierstone, E., Milman, P.D.: Arc-analytic functions. Invent. Math. 101, 411–424 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bierstone, E., Milman, P.D.: Relations among analytic functions, II. Ann. Inst. Fourier (Grenoble) 37(2), 49–77 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128, 207–302 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bierstone, E., Milman, P.D.: Resolution of singularities in Denjoy–Carleman classes. Selecta Math. (N.S.) 10, 1–28 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bierstone, E., Milman, P.D., Valette, G.: Arc-quasianalytic functions. Proc. Am. Math. Soc. 143, 3915–3925 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borel, E.: Sur la généralisation du prolongement analytique. C. R. Acad. Sci. Paris 130, 1115–1118 (1900)

    MATH  Google Scholar 

  9. Carleman, T.: Les Fonctions Quasi-Analytiques. Collection Borel, Gauthier-Villars (1926)

    MATH  Google Scholar 

  10. Chaumat, J., Chollet, A.-M.: Division par un polynôme hyperbolique. Can. J. Math. 56, 1121–1144 (2004)

    Article  MATH  Google Scholar 

  11. Childress, C.L.: Weierstrass division in quasianalytic local rings. Can. J. Math. 28, 938–953 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cluckers, R., Lipshitz, L.: Strictly convergent analytic structures. J. Euro. Math. Soc. 19, 107–149 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Denjoy, A.: Sur les fonctions quasi-analytiques de variable reélle. C. R. Acad. Sci. Paris 173, 3120–1322 (1921)

    MATH  Google Scholar 

  14. Glaeser, G.: Fonctions composées différentiables. Ann. Math. 77, 193–209 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grauert, H., Remmert, R.: Analytische Stellenalgebren, Grundlehren der Mathematischen Wissenschaften 176. Springer, Berlin (1971)

    Google Scholar 

  16. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale Univ. Press, New Haven (1923)

    MATH  Google Scholar 

  17. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983)

    MATH  Google Scholar 

  18. Komatsu, H.: The implicit function theorem for ultradifferentiable mappings. Proc. Jpn. Acad. 55, 69–72 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Langenbruch, M.: Extension of ultradifferentiable functions. Manuscr. Math. 83, 123–143 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Math, vol. 3. Oxford University Press, London (1966)

  21. Mandelbrojt, S.: Séries Adhérentes, Régularisation des Suites, Applications. Collection Borel, Gauthiers-Villars (1952)

    MATH  Google Scholar 

  22. Miller, C.: Infinite differentiability in polynomially bounded \(o\)-minimal structures. Proc. Am. Math. Soc. 123, 2551–2555 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nazarov, F., Sodin, M., Volberg, A.: Lower bounds for quasianalytic functions. I. How to control smooth functions. Math. Scand. 95, 59–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nowak, K.J.: A note on Bierstone-Milman-Pawłucki’s paper Composite differentiable functions. Ann. Polon. Math. 102, 293–299 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nowak, K.J.: On division of quasianalytic function germs. Int. J. Math. 13, 1–5 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Nowak, K.J.: Quantifier elimination in quasianalytic structures via non-standard analysis. Ann. Polon. Math. 114, 235–267 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rolin, J.-P., Speissegger, P., Wilkie, A.J.: Quasianalytic Denjoy–Carleman classes and o-minimality. J. Am. Math. Soc. 16, 751–777 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Roumieu, C.:Ultradistributions définies sur \({\rm R}^{\rm n}\) et sur certaines classes de variétés différentiables, J. Analyse Math. 10 (1962–63), 153–192

  29. Thilliez, V.: On quasianalytic local rings. Expo. Math. 26, 1–23 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Thilliez, On the non-extendability of quasianalytic germs, preprint (2010). arXiv:1006.4171v1 [mathCA]

Download references

Author information

Author notes

  1. André Belotto da Silva

    Present address: Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    André Belotto da Silva, Iwo Biborski & Edward Bierstone

Authors
  1. André Belotto da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iwo Biborski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward Bierstone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edward Bierstone.

Additional information

Research supported in part by NSERC Grant OGP0009070.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belotto da Silva, A., Biborski, I. & Bierstone, E. Solutions of quasianalytic equations. Sel. Math. New Ser. 23, 2523–2552 (2017). https://doi.org/10.1007/s00029-017-0345-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-017-0345-3

Keywords

Mathematics Subject Classification

Search

Navigation