An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms

José Cano

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 1, page 125-142
  • ISSN: 0373-0956

Abstract

top
We give a proof of the fact that any holomorphic Pfaffian form in two variables has a convergent integral curve. The proof gives an effective method to construct the solution, and we extend it to get a Gevrey type solution for a Gevrey form.

How to cite

top

Cano, José. "An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms." Annales de l'institut Fourier 43.1 (1993): 125-142. <http://eudml.org/doc/74984>.

@article{Cano1993,
abstract = {We give a proof of the fact that any holomorphic Pfaffian form in two variables has a convergent integral curve. The proof gives an effective method to construct the solution, and we extend it to get a Gevrey type solution for a Gevrey form.},
author = {Cano, José},
journal = {Annales de l'institut Fourier},
keywords = {Newton polygon; holomorphic; Pfaffian form; convergent integral curve; Gevrey type solution; Gevrey form},
language = {eng},
number = {1},
pages = {125-142},
publisher = {Association des Annales de l'Institut Fourier},
title = {An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms},
url = {http://eudml.org/doc/74984},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Cano, José
TI - An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 1
SP - 125
EP - 142
AB - We give a proof of the fact that any holomorphic Pfaffian form in two variables has a convergent integral curve. The proof gives an effective method to construct the solution, and we extend it to get a Gevrey type solution for a Gevrey form.
LA - eng
KW - Newton polygon; holomorphic; Pfaffian form; convergent integral curve; Gevrey type solution; Gevrey form
UR - http://eudml.org/doc/74984
ER -

References

top
  1. [1] I. BENDIXON, Sur les points singuliers des équations différentiels, Ofv. Kongl. Vetenskaps Akademiens Förhandlinger, Stokholm, 9 # 186 (1898), 635-658. JFM29.0275.01
  2. [2] C.A. BRIOT and J.C. BOUQUET, Propriétés des fonctions définies par des équations différentiels, Journal de l'Ecole Polytechnique, 36 (1856), 133-198. 
  3. [3] C. CAMACHO and P. SAD, Invariant varieties through singularities of holomorphic vector fields, Annals of Mathematics, 115 (1982), 579-595. Zbl0503.32007MR83m:58062
  4. [4] F. CANO, Desingularizations of plane vector fields, Transactions of the A.M.S., 296 (1986), 83-93. Zbl0612.14011MR87j:14009
  5. [5] J. CANO, On the series definied by differential equations, with an extension of the Puiseux Polygon construction to these equations, to appear in the International Mathematical Journal of Analysis and its Applications. Zbl0793.34009
  6. [6] H.B. FINE, On the Functions Definied by Differential Equations, with an Extension of the Puiseux Polygon Construction to these Equations, Amer. Jour. of Math., XI (1889), 317-328. Zbl21.0302.01JFM21.0302.01
  7. [7] H.B. FINE, Singular Solutions of Ordinary Differential Equations, Amer. Jour. of Math., XII (1890), 295-322. Zbl22.0302.02JFM22.0302.02
  8. [8] E.L. INCE, Ordinary Differential Equations, Dover Publications, 1926, 295-303. 
  9. [9] K. MAHLER, On formal power series as integrals of algebraic differential equations, Lincei-Rend. Sc. Fis. Mat. e Nat., L (1971), 76-89. Zbl0223.12107MR45 #8649
  10. [10] E. MAILLET, Sur les séries divergentes et les équations différentiels, Ann. Sci. École Norm. Sup., (1903), 487-518. Zbl34.0282.01JFM34.0282.01
  11. [11] B. MALGRANGE, Sur le théorème de Maillet, Asymptotic Anal. 2 (1989), 1-4. Zbl0693.34004MR90f:32005
  12. [12] J.F. MATTEI et R. MOUSSU, Holonomie et intégrales premières, Ann. Scient. Éc. Norm. Sup. 4ème serie, 13 (1980), 469-523. Zbl0458.32005MR83b:58005
  13. [13] H. PONCARÉ, Sur les propriétés des fonctions définies par les équations aux différences partielles, Thèse, Paris, 1879. 
  14. [14] J.P. RAMIS, Devissage Gevrey, Astérisque, 59/60 (1978), 173-204. Zbl0409.34018MR81g:34010
  15. [15] J.P. RAMIS, Théorèmes d'indices Gevrey pour les équations différentielles ordinaires, Memoirs of the American Mathematical Society, 296 (1984), 1-95. Zbl0555.47020MR86e:34021
  16. [16] J.F. RITT, On the singular solutions of algebraic differential equations, Ann. of Math., 37 (1936), 552-617. Zbl0015.25505JFM62.0518.01
  17. [17] A. SEIDENBERG, Reduction of singularities of the differential equation Ady = Bdx, Amer. J. Math., (1968), 248-269. Zbl0159.33303MR36 #3762
  18. [18] R.J. WALKER, Algebraic Curves, Dover Publications, 1962, 93-96. Zbl0103.38202

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.