Center conditions at infinity for Abel differential equations

Abstract

An Abel differential equation $y’=p(x)y^2 + q(x) y^3$ is said to have a center at a set $A=\{a_1,\dots,a_r\}$ of complex numbers if $y(a_1)=y(a_2)=\dots=y(a_r)$ for any solution $y(x)$ (with the initial value $y(a_1)$ small enough).

The polynomials $p,q$ are said to satisfy the “Polynomial Composition Condition” on $A$ if there exist polynomials $\tilde P$, $\tilde{Q}$ and $W$ such that $P=\int p$ and $Q=\int q$ are representable as $P(x)=\tilde{P}(W(x))$, $Q(x)=\tilde{Q}(W(x))$, and $W(a_1)=W(a_2)=…=W(a_r)$. We show that for wide ranges of degrees of $P$ and $Q$ (restricted only by certain assumptions on the common divisors of these degrees) the composition condition provides a very accurate approximation of the Center one — up to a finite number of configurations not accounted for. To our best knowledge, this is the first “general” (i.e., not restricted to small degrees of $p$ and $q$ or to a very special form of these polynomials) result in the Center problem for Abel equations.

As an important intermediate result we show that “at infinity” (according to an appropriate projectivization of the parameter space) the Center conditions are given by a system of the “Moment equations” of the form $\int^{a_s}_{a_1} P^k q = 0$, $s=2,\dots,r$, $k=0,1,…\; $.

  • [al] M. A. M. Alwash and N. G. Lloyd, "Nonautonomous equations related to polynomial two-dimensional systems," Proc. Roy. Soc. Edinburgh, vol. 105A, pp. 129-152, 1987.
    @article {al, MRKEY = {890049},
      AUTHOR = {Alwash, M. A. M. and Lloyd, N. G.},
      TITLE = {Nonautonomous equations related to polynomial two-dimensional systems},
      JOURNAL = {Proc. Roy. Soc. Edinburgh},
      FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics},
      VOLUME = {105A},
      YEAR = {1987},
      PAGES = {129--152},
      ISSN = {0308-2105},
      CODEN = {PEAMDU},
      MRCLASS = {58F21 (34A20 34C05 34C25 58F14)},
      MRNUMBER = {88i:58145},
      MRREVIEWER = {Jinghuang Tian},
      }
  • [alw1] M. A. M. Alwash, "On a condition for a centre of cubic nonautonomous equations," Proc. Roy. Soc. Edinburgh, vol. 113A, iss. 3-4, pp. 289-291, 1989.
    @article {alw1, MRKEY = {1037734},
      AUTHOR = {Alwash, M. A. M.},
      TITLE = {On a condition for a centre of cubic nonautonomous equations},
      JOURNAL = {Proc. Roy. Soc. Edinburgh},
      FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics},
      VOLUME = {113A},
      YEAR = {1989},
      NUMBER = {3-4},
      PAGES = {289--291},
      ISSN = {0308-2105},
      CODEN = {PEAMDU},
      MRCLASS = {34C05 (34A20)},
      MRNUMBER = {90m:34069},
      ZBLNUMBER={0685.34025},
      }
  • [alw2] M. A. M. Alwash, "Word problems and the centers of Abel differential equations," Ann. Differential Equations, vol. 11, iss. 4, pp. 392-396, 1995.
    @article {alw2, MRKEY = {1384574},
      AUTHOR = {Alwash, M. A. M.},
      TITLE = {Word problems and the centers of {A}bel differential equations},
      JOURNAL = {Ann. Differential Equations},
      FJOURNAL = {Annals of Differential Equations. Weifen Fangcheng Niankan},
      VOLUME = {11},
      YEAR = {1995},
      NUMBER = {4},
      PAGES = {392--396},
      ISSN = {1002-0942},
      MRCLASS = {34C05},
      MRNUMBER = {1384574},
      ZBLNUMBER = {0684.34025},
      }
  • [alw3] Go to document M. A. M. Alwash, "Periodic solutions of a quartic differential equation and Groebner bases," J. Comput. Appl. Math., vol. 75, iss. 1, pp. 67-76, 1996.
    @article {alw3, MRKEY = {1424887},
      AUTHOR = {Alwash, M. A. M.},
      TITLE = {Periodic solutions of a quartic differential equation and {G}roebner bases},
      JOURNAL = {J. Comput. Appl. Math.},
      FJOURNAL = {Journal of Computational and Applied Mathematics},
      VOLUME = {75},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {67--76},
      ISSN = {0377-0427},
      CODEN = {JCAMDI},
      MRCLASS = {34C25 (13P10 68Q40)},
      MRNUMBER = {97g:34055},
      DOI = {10.1016/S0377-0427(96)00059-3},
      ZBLNUMBER = {0864.34035},
      }
  • [alw4] Go to document M. A. M. Alwash, "On the center conditions of certain cubic systems," Proc. Amer. Math. Soc., vol. 126, iss. 11, pp. 3335-3336, 1998.
    @article {alw4, MRKEY = {1485453},
      AUTHOR = {Alwash, M. A. M.},
      TITLE = {On the center conditions of certain cubic systems},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {126},
      YEAR = {1998},
      NUMBER = {11},
      PAGES = {3335--3336},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {34C05},
      MRNUMBER = {99a:34073},
      MRREVIEWER = {Henryk {\.Z}o{\l}polhk adek},
      DOI = {10.1090/S0002-9939-98-04715-7},
      ZBLNUMBER = {0903.34026},
      }
  • [alw5] M. A. M. Alwash, "On the composition conjectures," Electron. J. Differential Equations, p. 69, 2003.
    @article {alw5, MRKEY = {1993777},
      AUTHOR = {Alwash, M. A. M.},
      TITLE = {On the composition conjectures},
      JOURNAL = {Electron. J. Differential Equations},
      FJOURNAL = {Electronic Journal of Differential Equations},
      YEAR = {2003},
      PAGES = {No. 69, 4 pp.},
      ISSN = {1072-6691},
      MRCLASS = {34C25},
      MRNUMBER = {2004f:34060},
      MRREVIEWER = {Bo Hua Yu},
      ZBLNUMBER={1054.34047},
      }
  • [ar] V. Arnol’d, "Problems on singularities and dynamical systems," in Developments in Mathematics: The Moscow School, London: Chapman & Hall, 1993, pp. 251-274.
    @incollection {ar, MRKEY = {1264427},
      AUTHOR = {Arnol'd, V.},
      TITLE = {Problems on singularities and dynamical systems},
      BOOKTITLE = {Developments in Mathematics: The {M}oscow School},
      EDITORS={Arnol'd, V. and Monastyrsky, M.},
      PAGES = {251--274},
      PUBLISHER = {Chapman \& Hall},
      ADDRESS = {London},
      YEAR = {1993},
      MRCLASS = {58C27 (53A05 57R45 58-02)},
      MRNUMBER = {95f:58015},
      MRREVIEWER = {D. R. J. Chillingworth},
      ZBLNUMBER = {0883.58016},
      }
  • [ari] V. Arnol’d and Y. Il’yashenko, "Ordinary differential equations," in Dynamical Systems. I, New York: Springer-Verlag, 1988, vol. 1, p. x.
    @incollection {ari, MRKEY = {970793},
      AUTHOR={Arnol'd, V. and Il'yashenko, Y.},
      TITLE={Ordinary differential equations},
      BOOKTITLE = {Dynamical Systems. {\rm I}},
      SERIES = {Encycl. Math. Sci.},
      VOLUME = {1},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1988},
      PAGES = {x+233},
      ISBN = {3-540-17000-6},
      MRCLASS = {58Fxx (34Cxx)},
      MRNUMBER = {89g:58060},
      ZBLNUMBER = {0471.70025},
      }
  • [bli1] M. Blinov, "Some computations around the center problem, related to the composition algebra of univariate polynomials," PhD Thesis , Weizmann Institute of Science, 1997.
    @phdthesis{bli1,
      author = {Blinov, M.},
      TITLE={Some computations around the center problem, related to the composition algebra of univariate polynomials},
      TYPE={M.Sc. Thesis},
      SCHOOL={Weizmann Institute of Science},
      YEAR={1997},
      }
  • [bli2] M. Blinov, "Center and Composition conditions for Abel Equation," PhD Thesis , Weizmann Institute of Science, 2002.
    @phdthesis{bli2,
      author = {Blinov, M.},
      TITLE={Center and Composition conditions for Abel Equation},
      SCHOOL={Weizmann Institute of Science},
      YEAR={2002},
      }
  • [bby] M. Blinov, M. Briskin, and Y. Yomdin, "Local center conditions for the Abel equation and cyclicity of its zero solution," in Complex Analysis and Dynamical Systems II, Providence, RI: Amer. Math. Soc., 2005, vol. 382, pp. 65-82.
    @incollection {bby, MRKEY = {2175877},
      AUTHOR = {Blinov, M. and Briskin, M. and Yomdin, Y.},
      TITLE = {Local center conditions for the {A}bel equation and cyclicity of its zero solution},
      BOOKTITLE = {Complex Analysis and Dynamical Systems {\rm II}},
      SERIES = {Contemp. Math.},
      VOLUME = {382},
      PAGES = {65--82},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      MRCLASS = {34C07 (34C25 37C27)},
      MRNUMBER = {2007b:34079},
      ZBLNUMBER = {1095.34018},
      }
  • [bri1] M. Briskin, "Infinitesimal center-focus problem, generalized moments and composition of polynomials," Funct. Differ. Equ., vol. 6, iss. 1-2, pp. 47-53, 1999.
    @article {bri1, MRKEY = {1733226},
      AUTHOR = {Briskin, M.},
      TITLE = {Infinitesimal center-focus problem, generalized moments and composition of polynomials},
      JOURNAL = {Funct. Differ. Equ.},
      FJOURNAL = {Functional Differential Equations},
      VOLUME = {6},
      YEAR = {1999},
      NUMBER = {1-2},
      PAGES = {47--53},
      ISSN = {0793-1786},
      MRCLASS = {34C05 (12D05 34C08)},
      MRNUMBER = {2000k:34049},
      MRREVIEWER = {I. D. Iliev},
      ZBLNUMBER = {1043.34025},
      }
  • [bri2] M. Briskin, Algebra of generalized moments and composition of polynomials, 2000.
    @misc{bri2,
      author = {Briskin, M.},
      TITLE = {Algebra of generalized moments and composition of polynomials},
      NOTE={Proc. Int. Conference ``Dynamical Systems: Stability, Control, Optimization'', Minsk},
      YEAR={2000},
      }
  • [bri3] M. Briskin, Recursive moments representation and quantitative moment problem.
    @misc{bri3,
      author = {Briskin, M.},
      TITLE = {Recursive moments representation and quantitative moment problem},
      NOTE={in preparation},
      }
  • [bfy1] Go to document M. Briskin, J. -P. Françoise, and Y. Yomdin, "Une approche au problème du centre-foyer de Poincaré," C. R. Acad. Sci. Paris Sér. I Math., vol. 326, iss. 11, pp. 1295-1298, 1998.
    @article {bfy1, MRKEY = {1649140},
      AUTHOR = {Briskin, M. and Fran{ç}oise, J.-P. and Yomdin, Y.},
      TITLE = {Une approche au problème du centre-foyer de {P}oincaré},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {326},
      YEAR = {1998},
      NUMBER = {11},
      PAGES = {1295--1298},
      ISSN = {0764-4442},
      CODEN = {CASMEI},
      MRCLASS = {34C05 (34C29 34E05 58F21)},
      MRNUMBER = {99j:34030},
      MRREVIEWER = {Robert Roussarie},
      DOI = {10.1016/S0764-4442(98)80182-1},
      ZBLNUMBER = {0914.34025},
      }
  • [bfy2] Go to document M. Briskin, J. -P. Françoise, and Y. Yomdin, "Center conditions, compositions of polynomials and moments on algebraic curves," Ergodic Theory Dynam. Systems, vol. 19, iss. 5, pp. 1201-1220, 1999.
    @article {bfy2, MRKEY = {1721616},
      AUTHOR = {Briskin, M. and Fran{ç}oise, J.-P. and Yomdin, Y.},
      TITLE = {Center conditions, compositions of polynomials and moments on algebraic curves},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {19},
      YEAR = {1999},
      NUMBER = {5},
      PAGES = {1201--1220},
      ISSN = {0143-3857},
      MRCLASS = {34C05 (34A25 34C08)},
      MRNUMBER = {2000k:34051},
      MRREVIEWER = {Lubomir Gavrilov},
      DOI = {10.1017/S0143385799141737},
      ZBLNUMBER = {0990.34017},
      }
  • [bfy3] Go to document M. Briskin, J. -P. Françoise, and Y. Yomdin, "Center conditions. II. Parametric and model center problems," Israel J. Math., vol. 118, pp. 61-82, 2000.
    @article {bfy3, MRKEY = {1776076},
      AUTHOR = {Briskin, M. and Fran{ç}oise, J.-P. and Yomdin, Y.},
      TITLE = {Center conditions. {II}. {P}arametric and model center problems},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {118},
      YEAR = {2000},
      PAGES = {61--82},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {34C05 (34C25)},
      MRNUMBER = {2001j:34033},
      MRREVIEWER = {Lubomir Gavrilov},
      DOI = {10.1007/BF02803516},
      ZBLNUMBER={0989.34021},
      }
  • [bfy4] Go to document M. Briskin, J. -P. Françoise, and Y. Yomdin, "Center conditions. III. Parametric and model center problems," Israel J. Math., vol. 118, pp. 83-108, 2000.
    @article {bfy4, MRKEY = {1776077},
      AUTHOR = {Briskin, M. and Fran{ç}oise, J.-P. and Yomdin, Y.},
      TITLE = {Center conditions. {III}. {P}arametric and model center problems},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {118},
      YEAR = {2000},
      PAGES = {83--108},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {34C05 (34C25)},
      MRNUMBER = {2001j:34034},
      MRREVIEWER = {Lubomir Gavrilov},
      DOI = {10.1007/BF02803517},
      ZBLNUMBER={0989.34022},
      }
  • [bfy5] M. Briskin, J. -P. Françoise, and Y. Yomdin, "Generalized moments, center-focus conditions, and compositions of polynomials," in Operator Theory, System Theory and Related Topics, Basel: Birkhäuser, 2001, pp. 161-185.
    @incollection {bfy5, MRKEY = {1821911},
      AUTHOR = {Briskin, M. and Fran{ç}oise, J.-P. and Yomdin, Y.},
      TITLE = {Generalized moments, center-focus conditions, and compositions of polynomials},
      BOOKTITLE = {Operator Theory, System Theory and Related Topics},
      VENUE={{B}eer-{S}heva/{R}ehovot, 1997},
      SERIES = {Oper. Theory Adv. Appl.},
      NUMBER = {123},
      PAGES = {161--185},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2001},
      MRCLASS = {34C05 (34C23 34C25)},
      MRNUMBER = {2002e:34050},
      MRREVIEWER = {Lubomir Gavrilov},
      ZBLNUMBER = {1075.34509},
      }
  • [by] M. Briskin and Y. Yomdin, "Tangential version of Hilbert 16th problem for the Abel equation," Mosc. Math. J., vol. 5, iss. 1, pp. 23-53, 2005.
    @article {by, MRKEY = {2153465},
      AUTHOR = {Briskin, M. and Yomdin, Y.},
      TITLE = {Tangential version of {H}ilbert 16th problem for the {A}bel equation},
      JOURNAL = {Mosc. Math. J.},
      FJOURNAL = {Moscow Mathematical Journal},
      VOLUME = {5},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {23--53},
      ISSN = {1609-3321},
      MRCLASS = {34C07 (34C08 37C10)},
      MRNUMBER = {2006c:34067},
      MRREVIEWER = {Yulin Zhao},
      ZBLNUMBER = {1097.34025},
      }
  • [bru1] Go to document A. Brudnyi, "An algebraic model for the center problem," Bull. Sci. Math., vol. 128, iss. 10, pp. 839-857, 2004.
    @article {bru1, MRKEY = {2100848},
      AUTHOR = {Brudnyi, Alexander},
      TITLE = {An algebraic model for the center problem},
      JOURNAL = {Bull. Sci. Math.},
      FJOURNAL = {Bulletin des Sciences Mathématiques},
      VOLUME = {128},
      YEAR = {2004},
      NUMBER = {10},
      PAGES = {839--857},
      ISSN = {0007-4497},
      CODEN = {BSMQA9},
      MRCLASS = {34C07 (37C10)},
      MRNUMBER = {2005k:34122},
      MRREVIEWER = {Manuel Sanchis},
      DOI = {10.1016/j.bulsci.2004.05.006},
      ZBLNUMBER = {1084.34035},
      }
  • [bru2] Go to document A. Brudnyi, "An explicit expression for the first return map in the center problem," J. Differential Equations, vol. 206, iss. 2, pp. 306-314, 2004.
    @article {bru2, MRKEY = {2095816},
      AUTHOR = {Brudnyi, Alexander},
      TITLE = {An explicit expression for the first return map in the center problem},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {206},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {306--314},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {34C07 (37C27)},
      MRNUMBER = {2005g:34067},
      MRREVIEWER = {Yulin Zhao},
      DOI = {10.1016/j.jde.2004.04.002},
      ZBLNUMBER = {1066.34026},
      }
  • [bru3] Go to document A. Brudnyi, "On the center problem for ordinary differential equations," Amer. J. Math., vol. 128, iss. 2, pp. 419-451, 2006.
    @article {bru3, MRKEY = {2214898},
      AUTHOR = {Brudnyi, Alexander},
      TITLE = {On the center problem for ordinary differential equations},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {128},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {419--451},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {34C25 (34C05 34C07 37C10)},
      MRNUMBER = {2007i:34064},
      MRREVIEWER = {M. A. Alwash},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.2brudnyi.pdf},
      ZBLNUMBER = {1104.34022},
      }
  • [bru4] Go to document A. Brudnyi, "On center sets of ODEs determined by moments of their coefficients," Bull. Sci. Math., vol. 130, iss. 1, pp. 33-48, 2006.
    @article {bru4, MRKEY = {2197179},
      AUTHOR = {Brudnyi, Alexander},
      TITLE = {On center sets of {ODE}s determined by moments of their coefficients},
      JOURNAL = {Bull. Sci. Math.},
      FJOURNAL = {Bulletin des Sciences Mathématiques},
      VOLUME = {130},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {33--48},
      ISSN = {0007-4497},
      CODEN = {BSMQA9},
      MRCLASS = {34C07 (32E30 37C10)},
      MRNUMBER = {2006m:34082},
      MRREVIEWER = {Timoteo Carletti},
      DOI = {10.1016/j.bulsci.2005.06.003},
      ZBLNUMBER = {1102.34019},
      }
  • [bru5] A. Brudnyi, Private communication.
    @misc{bru5,
      author = {Brudnyi, Alexander},
      TITLE={private communication},
      }
  • [ite] Go to document K. T. Chen, "Iterated path integrals," Bull. Amer. Math. Soc., vol. 83, iss. 5, pp. 831-879, 1977.
    @article {ite, MRKEY = {0454968},
      AUTHOR = {Chen, Kuo Tsai},
      TITLE = {Iterated path integrals},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {83},
      YEAR = {1977},
      NUMBER = {5},
      PAGES = {831--879},
      ISSN = {0002-9904},
      MRCLASS = {55D35 (58A99)},
      MRNUMBER = {56 \#13210},
      MRREVIEWER = {Jean-Michel Lemaire},
      DOI = {10.1090/S0002-9904-1977-14320-6},
      ZBLNUMBER = {0389.58001},
      }
  • [che1] L. A. vCerkas, "The number of limit cycles of a certain second order autonumous system," Differencial’ nye Uravnenija, vol. 12, iss. 5, pp. 944-946, 960, 1976.
    @article {che1, MRKEY = {0422759},
      AUTHOR = {{\v{C}}erkas, L. A.},
      TITLE = {The number of limit cycles of a certain second order autonumous system},
      JOURNAL = {Differencial' nye Uravnenija},
      FJOURNAL = {Differentsial' nye Uravneniya},
      VOLUME = {12},
      YEAR = {1976},
      NUMBER = {5},
      PAGES = {944--946, 960},
      ISSN = {0374-0641},
      MRCLASS = {34C05},
      MRNUMBER = {54 \#10745},
      MRREVIEWER = {M. Bertolino},
      }
  • [chr1] Go to document C. Christopher, "An algebraic approach to the classification of centers in polynomial Liénard systems," J. Math. Anal. Appl., vol. 229, iss. 1, pp. 319-329, 1999.
    @article {chr1, MRKEY = {1664344},
      AUTHOR = {Christopher, Colin},
      TITLE = {An algebraic approach to the classification of centers in polynomial {L}iénard systems},
      JOURNAL = {J. Math. Anal. Appl.},
      FJOURNAL = {Journal of Mathematical Analysis and Applications},
      VOLUME = {229},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {319--329},
      ISSN = {0022-247X},
      CODEN = {JMANAK},
      MRCLASS = {34C05},
      MRNUMBER = {99k:34056},
      MRREVIEWER = {Pavao Marde{š}i{ć}},
      DOI = {10.1006/jmaa.1998.6175},
      ZBLNUMBER = {0921.34033},
      }
  • [chr2] Go to document C. Christopher, "Abel equations: composition conjectures and the model problem," Bull. London Math. Soc., vol. 32, iss. 3, pp. 332-338, 2000.
    @article {chr2, MRKEY = {1750484},
      AUTHOR = {Christopher, Colin},
      TITLE = {Abel equations: composition conjectures and the model problem},
      JOURNAL = {Bull. London Math. Soc.},
      FJOURNAL = {The Bulletin of the London Mathematical Society},
      VOLUME = {32},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {332--338},
      ISSN = {0024-6093},
      CODEN = {LMSBBT},
      MRCLASS = {34C05},
      MRNUMBER = {2001d:34047},
      MRREVIEWER = {Robert Roussarie},
      DOI = {10.1112/S0024609300006998},
      ZBLNUMBER = {1047.34019},
      }
  • [chr3] C. Christopher and C. Li, Limit Cycles of Differential Equations, Basel: Birkhäuser, 2007.
    @BOOK {chr3, MRKEY = {2325099},
      AUTHOR = {Christopher, Colin and Li, Chengzhi},
      TITLE = {Limit Cycles of Differential Equations},
      SERIES = {Adv. Courses in Math. {\rm CRM Barcelona}},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2007},
      PAGES = {viii+171},
      ISBN = {978-3-7643-8409-8},
      MRCLASS = {34-01 (34-02 34C05 34C07 37C10 37C27)},
      MRNUMBER = {2008d:34001},
      MRREVIEWER = {I. D. Iliev},
      ZBLNUMBER = {05162643},
      }
  • [col] C. B. Collins, "Conditions for a centre in a simple class of cubic systems," Differential Integral Equations, vol. 10, iss. 2, pp. 333-356, 1997.
    @article {col, MRKEY = {1424815},
      AUTHOR = {Collins, C. B.},
      TITLE = {Conditions for a centre in a simple class of cubic systems},
      JOURNAL = {Differential Integral Equations},
      FJOURNAL = {Differential and Integral Equations. An International Journal for Theory \& Applications},
      VOLUME = {10},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {333--356},
      ISSN = {0893-4983},
      MRCLASS = {34C05},
      MRNUMBER = {98c:34043},
      MRREVIEWER = {Anna Cima},
      ZBLNUMBER = {0894.34022},
      }
  • [dev1] Go to document J. Devlin, "Word problems related to periodic solutions of a nonautonomous system," Math. Proc. Cambridge Philos. Soc., vol. 108, iss. 1, pp. 127-151, 1990.
    @article {dev1, MRKEY = {1049766},
      AUTHOR = {Devlin, James},
      TITLE = {Word problems related to periodic solutions of a nonautonomous system},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {108},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {127--151},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {58F21 (05A99 34C05)},
      MRNUMBER = {91f:58073},
      MRREVIEWER = {Jian Ying Zhou},
      DOI = {10.1017/S0305004100069012},
      }
  • [dev2] Go to document J. Devlin, "Word problems related to derivatives of the displacement map," Math. Proc. Cambridge Philos. Soc., vol. 110, iss. 3, pp. 569-579, 1991.
    @article {dev2, MRKEY = {1120490},
      AUTHOR = {Devlin, James},
      TITLE = {Word problems related to derivatives of the displacement map},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {110},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {569--579},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {34C25},
      MRNUMBER = {92h:34081},
      MRREVIEWER = {Jaume Llibre},
      DOI = {10.1017/S0305004100070638},
      ZBLNUMBER = {0744.34032},
      }
  • [dlp] Go to document J. Devlin, N. G. Lloyd, and J. M. Pearson, "Cubic systems and Abel equations," J. Differential Equations, vol. 147, iss. 2, pp. 435-454, 1998.
    @article {dlp, MRKEY = {1633961},
      AUTHOR = {Devlin, James and Lloyd, N. G. and Pearson, J. M.},
      TITLE = {Cubic systems and {A}bel equations},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {147},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {435--454},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {34C05 (34C25)},
      MRNUMBER = {99f:34037},
      MRREVIEWER = {Natalia Borisovna Medvedeva},
      DOI = {10.1006/jdeq.1998.3420},
      ZBLNUMBER = {0911.34020},
      }
  • [jpf1] Go to document J. P. Francoise, "Successive derivatives of a first return map, application to the study of quadratic vector fields," Ergodic Th. Dynam. Syst., vol. 16, iss. 1, pp. 87-96, 1996.
    @article {jpf1, MRKEY = {1375128},
      AUTHOR = {Francoise, J. P.},
      TITLE = {Successive derivatives of a first return map, application to the study of quadratic vector fields},
      JOURNAL = {Ergodic Th. Dynam. Syst.},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {16},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {87--96},
      ISSN = {0143-3857},
      MRCLASS = {58F14 (34C05 58F21)},
      MRNUMBER = {97a:58131},
      MRREVIEWER = {Robert Roussarie},
      DOI = {10.1017/S0143385700008725},
      ZBLNUMBER = {0852.34008},
      }
  • [jpf2] J. -P. Françoise, "Local bifurcations of limit cycles, Abel equations and Liénard systems," in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Dordrecht: Kluwer Acad. Publ., 2004, pp. 187-209.
    @incollection {jpf2, MRKEY = {2083247},
      AUTHOR = {Fran{ç}oise, J.-P.},
      TITLE = {Local bifurcations of limit cycles, {A}bel equations and {L}iénard systems},
      BOOKTITLE = {Normal Forms, Bifurcations and Finiteness Problems in Differential Equations},
      SERIES = {NATO Sci. Ser. {\rm II} Math. Phys. Chem.},
      NUMBER = {137},
      PAGES = {187--209},
      PUBLISHER = {Kluwer Acad. Publ.},
      ADDRESS = {Dordrecht},
      YEAR = {2004},
      MRCLASS = {34C05 (34C23 34E15)},
      MRNUMBER = {2005i:34034},
      MRREVIEWER = {A. P. Sadovski{\u\i}},
      }
  • [fy] Go to document J. -P. Francoise and Y. Yomdin, "Bernstein inequalities and applications to analytic geometry and differential equations," J. Funct. Anal., vol. 146, iss. 1, pp. 185-205, 1997.
    @article {fy, MRKEY = {1446379},
      AUTHOR = {Francoise, J.-P. and Yomdin, Y.},
      TITLE = {Bernstein inequalities and applications to analytic geometry and differential equations},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {146},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {185--205},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {34A25 (12H20 14P05 32B05 34A20 34C05)},
      MRNUMBER = {98h:34009c},
      MRREVIEWER = {Pierre D. Milman},
      DOI = {10.1006/jfan.1996.3029},
      ZBLNUMBER = {0869.34008},
      }
  • [gl] Go to document A. Gasull and J. Llibre, "Limit cycles for a class of Abel equations," SIAM J. Math. Anal., vol. 21, iss. 5, pp. 1235-1244, 1990.
    @article {gl, MRKEY = {1062402},
      AUTHOR = {Gasull, A. and Llibre, J.},
      TITLE = {Limit cycles for a class of {A}bel equations},
      JOURNAL = {{\rm SIAM} J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {21},
      YEAR = {1990},
      NUMBER = {5},
      PAGES = {1235--1244},
      ISSN = {0036-1410},
      CODEN = {SJMAAH},
      MRCLASS = {34C05 (34C25 58F21)},
      MRNUMBER = {91e:34036},
      MRREVIEWER = {E. J. Barbeau},
      DOI = {10.1137/0521068},
      ZBLNUMBER = {0732.34025},
      }
  • [gav] Go to document L. Gavrilov, "Higher order Poincaré-Pontryagin functions and iterated path integrals," Ann. Fac. Sci. Toulouse Math., vol. 14, iss. 4, pp. 663-682, 2005.
    @article {gav, MRKEY = {2188587},
      AUTHOR = {Gavrilov, Lubomir},
      TITLE = {Higher order {P}oincaré-{P}ontryagin functions and iterated path integrals},
      JOURNAL = {Ann. Fac. Sci. Toulouse Math.},
      FJOURNAL = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
      VOLUME = {14},
      YEAR = {2005},
      NUMBER = {4},
      PAGES = {663--682},
      ISSN = {0240-2963},
      MRCLASS = {34C07 (32S65 37C85 37F75)},
      MRNUMBER = {2006i:34074},
      MRREVIEWER = {Marc Chaperon},
      URL = {http://afst.cedram.org/item?id=AFST_2005_6_14_4_663_0},
      ZBLNUMBER = {1104.34024},
      }
  • [gi1] Go to document L. Gavrilov and I. D. Iliev, "Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems," Ergodic Theory Dynam. Systems, vol. 20, iss. 6, pp. 1671-1686, 2000.
    @article {gi1, MRKEY = {1804952},
      AUTHOR = {Gavrilov, Lubomir and Iliev, Iliya D.},
      TITLE = {Second-order analysis in polynomially perturbed reversible quadratic {H}amiltonian systems},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {20},
      YEAR = {2000},
      NUMBER = {6},
      PAGES = {1671--1686},
      ISSN = {0143-3857},
      MRCLASS = {34C05 (34C07 37J99)},
      MRNUMBER = {2003c:34029},
      MRREVIEWER = {Pavao Marde{š}i{ć}},
      DOI = {10.1017/S0143385700000936},
      ZBLNUMBER = {0992.37054},
      }
  • [gi2] Go to document L. Gavrilov and I. D. Iliev, "The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields," Amer. J. Math., vol. 127, iss. 6, pp. 1153-1190, 2005.
    @article {gi2, MRKEY = {2183522},
      AUTHOR = {Gavrilov, Lubomir and Iliev, Iliya D.},
      TITLE = {The displacement map associated to polynomial unfoldings of planar {H}amiltonian vector fields},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {127},
      YEAR = {2005},
      NUMBER = {6},
      PAGES = {1153--1190},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {34C07 (34C08 37C10 37C27 58K10)},
      MRNUMBER = {2006h:34065},
      MRREVIEWER = {Chun-Gen Liu},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v127/127.6gavrilov.pdf},
      ZBLNUMBER = {1093.34015},
      }
  • [gin] Go to document J. Giné, "The nondegenerate center problem and the inverse integrating factor," Bull. Sci. Math., vol. 130, iss. 2, pp. 152-161, 2006.
    @article {gin, MRKEY = {2200643},
      AUTHOR = {Gin{é},
      Jaume},
      TITLE = {The nondegenerate center problem and the inverse integrating factor},
      JOURNAL = {Bull. Sci. Math.},
      FJOURNAL = {Bulletin des Sciences Mathématiques},
      VOLUME = {130},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {152--161},
      ISSN = {0007-4497},
      CODEN = {BSMQA9},
      MRCLASS = {34A05 (34A25 34C05 37C10)},
      MRNUMBER = {2006i:34003},
      MRREVIEWER = {Anthony D. Osborne},
      DOI = {10.1016/j.bulsci.2005.09.001},
      ZBLNUMBER = {1102.34001},
      }
  • [gg] V. Golubitski and V. Guillemin, "Stable mappings and their singularities," Grad. Texts in Math., vol. 14, 1973.
    @article{gg,
      author={Golubitski, V. and Guillemin, V.},
      TITLE={Stable mappings and their singularities},
      JOURNAL={Grad. Texts in Math.},
      VOLUME={14},
      YEAR={1973},
      PAGES={NEED},
      }
  • [i] Go to document Y. Ilyashenko, "Centennial history of Hilbert’s 16th problem," Bull. Amer. Math. Soc., vol. 39, iss. 3, pp. 301-354, 2002.
    @article {i, MRKEY = {1898209},
      AUTHOR = {Ilyashenko, Yu.},
      TITLE = {Centennial history of {H}ilbert's 16th problem},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {39},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {301--354},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {34-02 (34C07 37C10 37F75)},
      MRNUMBER = {2003c:34001},
      MRREVIEWER = {Lubomir Gavrilov},
      DOI = {10.1090/S0273-0979-02-00946-1},
      ZBLNUMBER = {1004.34017},
      }
  • [iva.vol] V. V. Ivanov and E. P. Volokitin, "Uniformly isochronous polynomial centers," Electron. J. Differential Equations, p. 133, 2005.
    @article {iva.vol, MRKEY = {2181277},
      AUTHOR = {Ivanov, Vladimir V. and Volokitin, Evgenii P.},
      TITLE = {Uniformly isochronous polynomial centers},
      JOURNAL = {Electron. J. Differential Equations},
      FJOURNAL = {Electronic Journal of Differential Equations},
      YEAR = {2005},
      PAGES = {No. 133, 10 pp.},
      ISSN = {1072-6691},
      MRCLASS = {34C05 (34C25)},
      MRNUMBER = {2006g:34055},
      MRREVIEWER = {Isaac A. Garc{\'ı}a},
      }
  • [ln] Go to document A. Lins Neto, "On the number of solutions of the equation $dx/dt=\sum ^{n}_{j=0}\,a_{j}(t)x^{j}$, $0\leq t\leq 1$, for which $x(0)=x(1)$," Invent. Math., vol. 59, iss. 1, pp. 67-76, 1980.
    @article {ln, MRKEY = {575082},
      AUTHOR = {Lins Neto, Alcides},
      TITLE = {On the number of solutions of the equation {$dx/dt=\sum \sp{n}\sb{j=0}\,a\sb{j}(t)x\sp{j}$},
      {$0\leq t\leq 1$},
      for which {$x(0)=x(1)$}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {59},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {67--76},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {34B05},
      MRNUMBER = {81i:34009},
      MRREVIEWER = {D. Izyumova},
      DOI = {10.1007/BF01390315},
      ZBLNUMBER = {0448.34012},
      }
  • [llo] Go to document N. G. Lloyd, "The number of periodic solutions of the equation $\dot z=z^{N}+p_{1}(t)z^{N-1}+…+p_{N}(t)$," Proc. London Math. Soc., vol. 27, pp. 667-700, 1973.
    @article {llo, MRKEY = {0333356},
      AUTHOR = {Lloyd, N. G.},
      TITLE = {The number of periodic solutions of the equation {$\dot z=z\sp{N}+p\sb{1}(t)z\sp{N-1}+...+p\sb{N}(t)$}},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {27},
      YEAR = {1973},
      PAGES = {667--700},
      ISSN = {0024-6115},
      MRCLASS = {34C25},
      MRNUMBER = {48 \#11681},
      MRREVIEWER = {G. R. Sell},
      DOI = {10.1112/plms/s3-27.4.667},
      ZBLNUMBER = {0273.34042},
      }
  • [mov] Go to document H. Movasati, "Center conditions: rigidity of logarithmic differential equations," J. Differential Equations, vol. 197, iss. 1, pp. 197-217, 2004.
    @article {mov, MRKEY = {2030154},
      AUTHOR = {Movasati, Hossein},
      TITLE = {Center conditions: rigidity of logarithmic differential equations},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {197},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {197--217},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {34C07 (14D05 32S40 37F75)},
      MRNUMBER = {2005f:34080},
      MRREVIEWER = {Jorge Vit{ó}rio Pereira},
      DOI = {10.1016/j.jde.2003.07.002},
      ZBLNUMBER = {1049.32033},
      }
  • [pak5] Go to document F. Pakovich and M. Muzychuk, "Solution of the polynomial moment problem," Proc. Lond. Math. Soc., vol. 99, iss. 3, pp. 633-657, 2009.
    @article {pak5, MRKEY = {2551466},
      AUTHOR = {Pakovich, F. and Muzychuk, M.},
      TITLE = {Solution of the polynomial moment problem},
      JOURNAL = {Proc. Lond. Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {99},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {633--657},
      ISSN = {0024-6115},
      MRCLASS = {30E05 (12F10)},
      MRNUMBER = {2551466},
      DOI = {10.1112/plms/pdp010},
      ZBLNUMBER = {1177.30046},
      }
  • [pak1] Go to document F. Pakovich, "A counterexample to the “composition conjecture”," Proc. Amer. Math. Soc., vol. 130, iss. 12, pp. 3747-3749, 2002.
    @article {pak1, MRKEY = {1920083},
      AUTHOR = {Pakovich, F.},
      TITLE = {A counterexample to the ``composition conjecture''},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {130},
      YEAR = {2002},
      NUMBER = {12},
      PAGES = {3747--3749},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {30E05 (30C10 30D05 34C05)},
      MRNUMBER = {2003j:30053},
      MRREVIEWER = {Themistocles M. Rassias},
      DOI = {10.1090/S0002-9939-02-06755-2},
      }
  • [pak2] F. Pakovich, "On the polynomial moment problem," Math. Res. Lett., vol. 10, iss. 2-3, pp. 401-410, 2003.
    @article {pak2, MRKEY = {1983032},
      AUTHOR = {Pakovich, F.},
      TITLE = {On the polynomial moment problem},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {10},
      YEAR = {2003},
      NUMBER = {2-3},
      PAGES = {401--410},
      ISSN = {1073-2780},
      MRCLASS = {30C10 (30E05 44A60)},
      MRNUMBER = {2004c:30006},
      MRREVIEWER = {Olav Nj{\aa}stad},
      ZBLNUMBER = {1043.30021},
      }
  • [pak3] Go to document F. Pakovich, "On polynomials orthogonal to all powers of a Chebyshev polynomial on a segment," Israel J. Math., vol. 142, pp. 273-283, 2004.
    @article {pak3, MRKEY = {2085720},
      AUTHOR = {Pakovich, F.},
      TITLE = {On polynomials orthogonal to all powers of a {C}hebyshev polynomial on a segment},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {142},
      YEAR = {2004},
      PAGES = {273--283},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {33C45 (34M55 42C05)},
      MRNUMBER = {2005f:33017},
      MRREVIEWER = {Lubomir Gavrilov},
      DOI = {10.1007/BF02771537},
      ZBLNUMBER = {1058.30007},
      }
  • [pak4] Go to document F. Pakovich, "On polynomials orthogonal to all powers of a given polynomial on a segment," Bull. Sci. Math., vol. 129, iss. 9, pp. 749-774, 2005.
    @article {pak4, MRKEY = {2172140},
      AUTHOR = {Pakovich, F.},
      TITLE = {On polynomials orthogonal to all powers of a given polynomial on a segment},
      JOURNAL = {Bull. Sci. Math.},
      FJOURNAL = {Bulletin des Sciences Mathématiques},
      VOLUME = {129},
      YEAR = {2005},
      NUMBER = {9},
      PAGES = {749--774},
      ISSN = {0007-4497},
      CODEN = {BSMQA9},
      MRCLASS = {30E20 (30C10 34C07)},
      MRNUMBER = {2007d:30028},
      MRREVIEWER = {Stephen M. Zemyan},
      DOI = {10.1016/j.bulsci.2005.06.001},
      ZBLNUMBER = {1089.30039},
      }
  • [pry] Go to document F. Pakovich, N. Roytvarf, and Y. Yomdin, "Cauchy-type integrals of algebraic functions," Israel J. Math., vol. 144, pp. 221-291, 2004.
    @article {pry, MRKEY = {2121542},
      AUTHOR = {Pakovich, F. and Roytvarf, N. and Yomdin, Y.},
      TITLE = {Cauchy-type integrals of algebraic functions},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {144},
      YEAR = {2004},
      PAGES = {221--291},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {30E20 (34C07)},
      MRNUMBER = {2006e:30043},
      MRREVIEWER = {Stephen M. Zemyan},
      DOI = {10.1007/BF02916714},
      ZBLNUMBER = {1078.30034},
      }
  • [plc] Go to document J. M. Pearson, N. G. Lloyd, and C. J. Christopher, "Algorithmic derivation of centre conditions," SIAM Rev., vol. 38, iss. 4, pp. 619-636, 1996.
    @article {plc, MRKEY = {1420840},
      AUTHOR = {Pearson, J. M. and Lloyd, N. G. and Christopher, C. J.},
      TITLE = {Algorithmic derivation of centre conditions},
      JOURNAL = {SIAM Rev.},
      FJOURNAL = {SIAM Review. A Publication of the Society for Industrial and Applied Mathematics},
      VOLUME = {38},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {619--636},
      ISSN = {0036-1445},
      CODEN = {SIREAD},
      MRCLASS = {34C05 (34-04 58F21)},
      MRNUMBER = {98c:34050},
      MRREVIEWER = {Henryk {\.Z}o{\l}polhk adek},
      DOI = {10.1137/S0036144595283575},
      ZBLNUMBER = {0876.34033},
      }
  • [poi] A. Poincaré, Oeuvres, Les Grands Classiques, Villars: Gauthier, 1995.
    @book{poi,
      author={Poincaré, A.},
      TITLE={Oeuvres, Les Grands Classiques},
      PUBLISHER={Gauthier},
      ADDRESS={Villars},
      YEAR={1995},
      }
  • [rit] Go to document J. F. Ritt, "Prime and composite polynomials," Trans. Amer. Math. Soc., vol. 23, iss. 1, pp. 51-66, 1922.
    @article {rit, MRKEY = {1501189},
      AUTHOR = {Ritt, J. F.},
      TITLE = {Prime and composite polynomials},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {23},
      YEAR = {1922},
      NUMBER = {1},
      PAGES = {51--66},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {13B25},
      MRNUMBER = {1501189},
      DOI = {10.2307/1988911},
      }
  • [rou2] R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, Basel: Birkhäuser, 1998, vol. 164.
    @book {rou2, MRKEY = {1628014},
      AUTHOR = {Roussarie, Robert},
      TITLE = {Bifurcation of Planar Vector Fields and {H}ilbert's Sixteenth Problem},
      SERIES = {Prog. Math.},
      VOLUME= {164},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1998},
      PAGES = {xviii+204},
      ISBN = {3-7643-5900-5},
      MRCLASS = {58F14 (34C05 34C23 58F21)},
      MRNUMBER = {99k:58129},
      MRREVIEWER = {Sergei Yakovenko},
      ZBLNUMBER = {0898.58039},
      }
  • [ro] N. Roytvarf, "Generalized moments, composition of polynomials and Bernstein classes," in Entire Functions in Modern Analysis, Ramat Gan: Bar-Ilan Univ., 2001, pp. 339-355.
    @incollection {ro, MRKEY = {1890547},
      AUTHOR = {Roytvarf, N.},
      TITLE = {Generalized moments, composition of polynomials and {B}ernstein classes},
      BOOKTITLE = {Entire Functions in Modern Analysis},
      VENUE={{T}el-{A}viv, 1997},
      SERIES = {Israel Math. Conf. Proc.},
      NUMBER = {15},
      PAGES = {339--355},
      PUBLISHER = {Bar-Ilan Univ.},
      ADDRESS = {Ramat Gan},
      YEAR = {2001},
      MRCLASS = {30E05 (30C10 30D05 34C05)},
      MRNUMBER = {2003g:30068},
      MRREVIEWER = {Dmitri Novikov},
      ZBLNUMBER = {0996.30025},
      }
  • [ry] Go to document N. Roytwarf and Y. Yomdin, "Bernstein classes," Ann. Inst. Fourier $($Grenoble$)$, vol. 47, iss. 3, pp. 825-858, 1997.
    @article {ry, MRKEY = {1465788},
      AUTHOR = {Roytwarf, N. and Yomdin, Y.},
      TITLE = {Bernstein classes},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {47},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {825--858},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {34A25 (12H20 14H20 14P05 30B10 30C55 34A20 34C05)},
      MRNUMBER = {98h:34009a},
      MRREVIEWER = {Pierre D. Milman},
      URL = {http://www.numdam.org/item?id=AIF_1997__47_3_825_0},
      ZBLNUMBER = {0974.30524},
      }
  • [ry1] N. Roytvarf and Y. Yomdin, "Analytic continuation of Cauchy-type integrals," Funct. Differ. Equ., vol. 12, iss. 3-4, pp. 375-388, 2005.
    @article {ry1, MRKEY = {2137518},
      AUTHOR = {Roytvarf, N. and Yomdin, Y.},
      TITLE = {Analytic continuation of {C}auchy-type integrals},
      JOURNAL = {Funct. Differ. Equ.},
      FJOURNAL = {Functional Differential Equations},
      VOLUME = {12},
      YEAR = {2005},
      NUMBER = {3-4},
      PAGES = {375--388},
      ISSN = {0793-1786},
      MRCLASS = {30E20 (30D30)},
      MRNUMBER = {2006h:30028},
      MRREVIEWER = {D. M. Israfilov},
      ZBLNUMBER = {1072.30029},
      }
  • [sha] Go to document S. Shahshahani, "Periodic solutions of polynomial first order differential equations," Nonlinear Anal., vol. 5, iss. 2, pp. 157-165, 1981.
    @article {sha, MRKEY = {606697},
      AUTHOR = {Shahshahani, S.},
      TITLE = {Periodic solutions of polynomial first order differential equations},
      JOURNAL = {Nonlinear Anal.},
      FJOURNAL = {Nonlinear Analysis},
      VOLUME = {5},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {157--165},
      ISSN = {0362-546X},
      CODEN = {NOANDD},
      MRCLASS = {34C25},
      MRNUMBER = {82d:34052},
      MRREVIEWER = {G. R. Sell},
      DOI = {10.1016/0362-546X(81)90041-9},
      ZBLNUMBER = {0449.34027},
      }
  • [sch] Go to document A. Schinzel, Polynomials with Special Regard to Reducibility, Cambridge: Cambridge Univ. Press, 2000.
    @book {sch, MRKEY = {1770638},
      AUTHOR = {Schinzel, A.},
      TITLE = {Polynomials with Special Regard to Reducibility},
      SERIES = {Encyclopedia of Math. Appl.},
      NUMBER = {77},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2000},
      PAGES = {x+558},
      ISBN = {0-521-66225-7},
      MRCLASS = {11R09 (11C08 12D05 12E05 12E25)},
      MRNUMBER = {2001h:11135},
      MRREVIEWER = {Art{ū}ras Dubickas},
      DOI = {10.1017/CBO9780511542916},
      ZBLNUMBER = {0956.12001},
      }
  • [schl] Go to document D. Schlomiuk, "Algebraic particular integrals, integrability and the problem of the center," Trans. Amer. Math. Soc., vol. 338, iss. 2, pp. 799-841, 1993.
    @article {schl, MRKEY = {1106193},
      AUTHOR = {Schlomiuk, Dana},
      TITLE = {Algebraic particular integrals, integrability and the problem of the center},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {338},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {799--841},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {34C05 (58F14)},
      MRNUMBER = {93j:34037},
      MRREVIEWER = {E. J. Barbeau},
      DOI = {10.2307/2154430},
      ZBLNUMBER = {0777.58028},
      }
  • [sib] K. S. Sibirsky, Introduction to the Algebraic Theory of Invariants of Differential Equations, Manchester: Manchester Univ. Press, 1988.
    @book {sib, MRKEY = {981596},
      AUTHOR = {Sibirsky, Konstantin Sergeevich},
      TITLE = {Introduction to the Algebraic Theory of Invariants of Differential Equations},
      SERIES = {Nonlinear Science: Theory and Applications},
      PUBLISHER = {Manchester Univ. Press},
      ADDRESS = {Manchester},
      YEAR = {1988},
      PAGES = {viii+169},
      ISBN = {0-7190-2669-5},
      MRCLASS = {34C20 (15A72 34-02 58F25)},
      MRNUMBER = {89k:34057},
      ZBLNUMBER = {0691.34032},
      }
  • [s] Go to document S. Smale, "Mathematical problems for the next century," Math. Intelligencer, vol. 20, iss. 2, pp. 7-15, 1998.
    @article {s, MRKEY = {1631413},
      AUTHOR = {Smale, Steve},
      TITLE = {Mathematical problems for the next century},
      JOURNAL = {Math. Intelligencer},
      FJOURNAL = {The Mathematical Intelligencer},
      VOLUME = {20},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {7--15},
      ISSN = {0343-6993},
      CODEN = {MAINDC},
      MRCLASS = {01A67 (00A05)},
      MRNUMBER = {99h:01033},
      DOI = {10.1007/BF03025291},
      ZBLNUMBER = {0947.01011},
      }
  • [yt] Go to document L. Yang and Y. Tang, "Some new results on Abel equations," J. Math. Anal. Appl., vol. 261, iss. 1, pp. 100-112, 2001.
    @article {yt, MRKEY = {1850959},
      AUTHOR = {Yang, Lijun and Tang, Yun},
      TITLE = {Some new results on {A}bel equations},
      JOURNAL = {J. Math. Anal. Appl.},
      FJOURNAL = {Journal of Mathematical Analysis and Applications},
      VOLUME = {261},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {100--112},
      ISSN = {0022-247X},
      CODEN = {JMANAK},
      MRCLASS = {34B15 (34C07 34E05)},
      MRNUMBER = {2002e:34033},
      MRREVIEWER = {Hans W. Volkmer},
      DOI = {10.1006/jmaa.2001.7478},
      ZBLNUMBER = {0995.34031},
      }
  • [y1] Y. Yomdin, "Global finiteness properties of analytic families and algebra of their Taylor coefficients," in The Arnoldfest, Providence, RI: Amer. Math. Soc., 1999, pp. 527-555.
    @incollection {y1, MRKEY = {1733591},
      AUTHOR = {Yomdin, Y.},
      TITLE = {Global finiteness properties of analytic families and algebra of their {T}aylor coefficients},
      BOOKTITLE = {The {A}rnoldfest},
      VENUE={{T}oronto, {ON},
      1997},
      SERIES = {Fields Inst. Commun.},
      NUMBER = {24},
      PAGES = {527--555},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1999},
      MRCLASS = {34C10 (30C15 34M10)},
      MRNUMBER = {2001i:34060},
      MRREVIEWER = {Wies{\l}aw Ple{\'s}niak},
      ZBLNUMBER = {0944.34003},
      }
  • [yp] Y. Yomdin, "The center problem for the Abel equation, compositions of functions, and moment conditions," Moscow Math. J., vol. 3, iss. 3, pp. 1167-1195, 2003.
    @article {yp, MRKEY = {2078578},
      AUTHOR = {Yomdin, Y.},
      TITLE = {The center problem for the {A}bel equation, compositions of functions, and moment conditions},
      NOTE = {With an addendum by F. Pakovich},
      JOURNAL = {Moscow Math. J.},
      FJOURNAL = {Moscow Mathematical Journal},
      VOLUME = {3},
      YEAR = {2003},
      NUMBER = {3},
      PAGES = {1167--1195},
      ISSN = {1609-3321},
      MRCLASS = {34C07 (30E05 34C05)},
      MRNUMBER = {2005f:34082},
      MRREVIEWER = {Valery G. Romanovski{\u\i}},
      ZBLNUMBER = {1086.34031},
      }
  • [yc] Y. Yomdin and G. Comte, Tame Geometry with Application in Smooth Analysis, New York: Springer-Verlag, 2004.
    @book {yc, MRKEY = {2041428},
      AUTHOR = {Yomdin, Yosef and Comte, Georges},
      TITLE = {Tame Geometry with Application in Smooth Analysis},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {1834},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      PAGES = {viii+186},
      ISBN = {3-540-20612-4},
      MRCLASS = {14P10 (26B15 32S15 37C99)},
      MRNUMBER = {2005i:14076},
      MRREVIEWER = {Lev Birbrair},
      ZBLNUMBER = {1076.14079},
      }
  • [z1] Go to document H. Zoladek, "The problem of center for resonant singular points of polynomial vector fields," J. Differential Equations, vol. 137, iss. 1, pp. 94-118, 1997.
    @article {z1, MRKEY = {1451537},
      AUTHOR = {Zoladek, Henryk},
      TITLE = {The problem of center for resonant singular points of polynomial vector fields},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {137},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {94--118},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {34C05},
      MRNUMBER = {98g:34058},
      MRREVIEWER = {Natalia Borisovna Medvedeva},
      DOI = {10.1006/jdeq.1997.3260},
      }
  • [z2] Go to document H. Zoladek, "Asymptotic properties of abelian integrals arising in quadratic systems," Bull. Belg. Math. Soc. Simon Stevin, vol. 7, iss. 2, pp. 265-276, 2000.
    @article {z2, MRKEY = {1771561},
      AUTHOR = {Zoladek, Henryk},
      TITLE = {Asymptotic properties of abelian integrals arising in quadratic systems},
      JOURNAL = {Bull. Belg. Math. Soc. Simon Stevin},
      FJOURNAL = {Bulletin of the Belgian Mathematical Society. Simon Stevin},
      VOLUME = {7},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {265--276},
      ISSN = {1370-1444},
      MRCLASS = {34C08 (34C05)},
      MRNUMBER = {2001d:34052},
      MRREVIEWER = {Robert Roussarie},
      URL = {http://projecteuclid.org/getRecord?id=euclid.bbms/1103055691},
      }
  • [z3] H. Zoladek, Private communication.
    @misc{z3,
      author = {Zoladek, Henryk},
      TITLE={private communication},
      }

Authors

Miriam Briskin

Jerusalem College of Engineering
Ramat Bet Hakerem
PO Box 3566
Jerusalem 91035
Israel

Nina Roytvarf

Department of Mathematics
The Weizmann Institute of Science
Rehovot 76100
Israel

Yosef Yomdin

Department of Mathematics
The Weizmann Institute of Science
Rehovot 76100
Israel