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Representations of three-variable polynomial

Source: Nordic Mathematical Contest 2018 Problem 4

April 10, 2018
algebrapolynomial

Problem Statement

Let f=f(x,y,z)f = f(x,y,z) be a polynomial in three variables xx, yy, zz such that f(w,w,w)=0f(w,w,w) = 0 for all w∈Rw \in \mathbb{R}. Show that there exist three polynomials AA, BB, CC in these same three variables such that A+B+C=0A + B + C = 0 and f(x,y,z)=A(x,y,z)⋅(x−y)+B(x,y,z)⋅(y−z)+C(x,y,z)⋅(z−x). f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). Is there any polynomial ff for which these AA, BB, CC are uniquely determined?
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