Then by Proposition 3, we may assume that the defining equation of C is given by [Z.sup.d] + [F.sub.d](X,Y) = 0 for some

homogeneous polynomial [F.sub.d](X, Y) of degree d and P = (0:0:1).

Seip, The Bohnenblust-Hille inequality for

homogeneous polynomials is hypercontractive, Ann.

Since [f.sub.n](x, y) is a bivariate

homogeneous polynomial of degree n, we obtain the points set {(a : b : 0)} at infinity by solving the corresponding univariate polynomial.

where [f.sup.1.sub.j](u, v) and [f.sup.2.sub.j](u, v) are

homogeneous polynomials in (u, v) of degree j,j = 2, 3,..., with coefficients in [R.sup.2] and Ker[pi], respectively.

Let the

homogeneous polynomial [R.sub.j]([y.sub.0],..., [y.sub.j]) [element of] C[[y.sub.0],..., [y.sub.j] ] of degree [2.sub.j-1] be defined by the equation

L[[H.sub.0]] = -mx[H.sub.0], together with the fact that [H.sub.0] is a

homogeneous polynomial with weight-degree n yields

[less than] [r.sub.k] = n, let P(r,d) be the submodule of R[[x.sub.1], ..., [x.sub.n]] consisting of

homogeneous polynomials P of degree d that are symmetric in [x.sub.i] and [x.sub.i+1] if [r.sub.j] [less than] i [less than] [r.sub.j+1] for some j.

Let P [member of] [P.sub.s] ([l.sub.p]([C.sup.n])) be an N-

homogeneous polynomial. If N < [p], then P [equivalent to] 0.

Let us first prove that the dimension of the space of

homogeneous polynomials in Q[X] of degree n satisfying (14) is at most equal to [2.sup.n-1].

A

homogeneous polynomial (of degree n) P : A [right arrow] B is said to be orthogonally-additive if P(x + y) = P (x) + P (y) whenever x, y [member of] A are orthogonally (i.e., [absolute value of x] [and] [absolute value of y] = 0).

Given a decomposable

homogeneous polynomial F([[functions of x].sub.1],...,[[functions of x].[sub.n]), we denote by L the set of all linear factors of F.

Given a

homogeneous polynomial F in C [x, y, z], in [4], a quasi-toric decomposition of F is a collection of

homogeneous polynomials f, g, h [member of] C[x, y, z] such that [f.sup.p] + [g.sup.q] + [hp.sup.q] F = 0, for two co-prime positive integers p,q > 1.