M. Blinov et Y. Yomdin, Generalized centre conditions and multiplicities for polynomial Abel equations of small degrees, NONLINEARIT, 12(4), 1999, pp. 1013-1028
We consider an Abel equation(*)y' = p(x)y(2) +q(x)y(3) with p(x), q (x)-pol
ynomials in x. A centre condition for this equation (closely related to the
classical centre condition for polynomial vector fields on the plane) is t
hat y(0) = y(0) equivalent to y(1) for any solution y(x). This condition is
given by the vanishing of all the Taylor coefficients v(k)(1) in the devel
opment y(x) = y(0) + Sigma(k=2)(infinity) v(k)(x)y(0)(k). Following Briskin
el al (Centre Conditions, Composition of Polynomials and Moments on Algebr
aic Curves to appear) we introduce periods of the equation (*) as those ome
ga epsilon C, for which y(0) equivalent to y(omega) for any solution y(x) o
f (*). The generalized centre conditions are conditions on p, q under which
given a(1),..., a(k) are (exactly all) the periods of (*). A new basis for
the ideals I-k = (v(2),...,v(k)) has been produced in Briskin et al (1998
The Bautin ideal of the Abel equation Nonlinenrity 10), defined by a linear
recurrence relation. Using this basis and a special representation of poly
nomials, we extend results of Briskin et al (Centre Conditions, Composition
of Polynomials and Moments on Algebraic Curves to appear), proving for sma
ll degrees of p and q a composition conjecture, as stated in Alwash and Llo
yd (1987 Nonautonomous equations related to polynomial two-dimensional syst
ems Proc. R. Soc. Edinburgh A 105 129-52), Briskin et al (Centre Conditions
, Composition of Polynomials and Moments on Algebraic Curves to appear), Br
iskin er al (Center Conditions II: Parametric and Model Centre Problems to
appear). In particular, this provides transparent generalized centre condit
ions in the cases considered. We also compute maximal possible multiplicity
of the zero solution of (*), extending the results of Alwash and Lloyd (19
87 Non-autonomous equations related to polynomial two-dimensional systems P
roc. R. Sec. Edinburgh A 105 129-52).