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2 i j Πn = {p(x,y)= aijx y }, i+Xj≤n with complex coefficients. We use the following well-known concept of the associate polynomial 2 i j (see section 10.2, [4]) . Let p ∈ Πn, i.e., p(x,y)= i+j≤n aijx y . Then the following trivariate homogeneous polynomial is calledP associated with p :

i j k p¯(x,y,z)= aijx y z . i+jX+k=n

Evidently we have that x y p¯(x,y,z)= znp , , for z 6= 0. (1.1)  z z  Also we have that p(x,y) =p ¯(x,y, 1). 2 It is easily seen that a polynomial p ∈ Πn is factorizable if and only if the associated polynomialp ¯ is factorizable. Moreover, we have that

p = p1p2 ⇔ p¯ =p ¯1p¯2.

To simplify notation, we shall use the same letter ℓ, say, to denote the bivariate polynomial of degree 1 and the line described by the equation ℓ(x,y) = 0, or the trivariate homogeneous polynomial of degree 1 and the line in the described by the equation ℓ(x,y,z) = 0. The following result follows from the fundamental theorem of algebra (see Theorem 10.8, [4]).

Theorem 1.1. Let p be a bivariate homogeneous polynomial of degree n : ˙ 2 p ∈ Πn. Then p can be factorized into linear polynomials:

n p = ℓi, Yi=1

2 where ℓi ∈ Π˙ 1.

2 2 The results for homogeneous polynomials

Theorem 2.1. Let p(x,y,z) and q(x,y,z) be linearly independent second de- gree homogeneous polynomi