On The Moment Bautin Index For Abel Equations

Abel equations in the form y'= p(x)y~2 + q(x)y~3 are of interests because of theirclose relation to the classical center-focus problem and Hilbert's 16th problem. In thispaper, we introduce a new way to explore Moment Bautin Index. An explicit lowerbound (we call it Quasi Moment Bautin Index) of Moment Bautin Index for generaldefinite polynomials is presented, this is our first main result. This Quasi MomentBautin Index depends on the structure of the generation of definite polynomial, andin particular, it coincides with Moment Bautin Index in the case of degree two. Forgeneral case, we give a classification of the generation of polynomials with degreeless than ten and their explicit Quasi Moment Bautin Index. Computing results ofMoment Bautin Index for definite polynomials P with small degree and small d ( dis a given number as the upper bound of the degree of polynomial q) are given byan algorithm based on Quasi Moment Bautin Index under particular parameters, andsurprisingly, these two indices coincide. Further, it can be proved that the parametersdo not in?uence the above result, therefore we can give Moment Bautin Index forsome particular polynomials, this is our second main result. Moreover, it's reasonableto conjecture that the two indices are the same essentially.