For a

homogeneous polynomial F(X, Y, z), the core of F(X, Y, Z) is defined as the sum of all terms of F(X, Y, Z) with the greatest exponent.

On the other side, we give the representation of quadratic polynomial in the case of real inner product as follows: [q.sub.2](z) = <Az,z), where q = [q.sub.1] + [q.sub.2], [q.sub.1] is linear, [q.sub.2] is a

homogeneous polynomial of degree 2, and A = [A.sub.nxn] is a complex matrix symmetric in the real sense.

If P is a

homogeneous polynomial of degree m on [K.sup.n] given by

for a nonconstant square-free

homogeneous polynomial F(x, y, z) [member of] C[x, y, z].

where [u.sub.i], i = 4, ..., 10 is a

homogeneous polynomial in x, y of degree i, respectively, (which can be seen in the Appendix); thus [??] and [??] in system (6) are two polynomials with degree 11.

TISSEUR, Perturbation theory for

homogeneous polynomial eigenvalue problems, Linear Algebra Appl., 358 (2003), pp.

is the third order terms of the equation which is obtained after computing the second order terms of the normal form, [U.sup.1.sub.2] (x, 0) the solution of equation [M.sup.1.sub.2][U.sup.1.sub.2] (x, 0) = [f.sup.1.sub.2] (x, 0, 0) and h = [([h.sup.1], [h.sup.2]).sup.T] is a second order

homogeneous polynomial in ([x.sub.1], [x.sub.2], [alpha]) with coefficients in [Q.sup.1].

Let furthermore D = {P = 0} be a divisor in [P.sub.n], given by a

homogeneous polynomial P.

An improvement on the number of limit cycles bifurcating from a nondegenerate center of

homogeneous polynomial systems was given in [16].

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a

homogeneous polynomial and the right-hand side is determined by the action of [A.sup.-1] on the column vector as a matrix multiplication.

Note that this is a graded linear map and that [[psi].sub.X] maps to zero any

homogeneous polynomial whose degree is at least N - d + 1.

Since [h.sub.n] must be a

homogeneous polynomial we must have [k.sub.1] = -m with m [member of] N [union]{0}, [k.sub.2] = [k.sub.3] = 0 and [C.sub.n] (y, z) [member of] C[y,z] \ {0}.