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4 three variable polynomials

Source: Iran 3rd rouund 2012-Final exam-P5

September 25, 2012
algebrapolynomialinvariantalgebra proposed

Problem Statement

We call the three variable polynomial PP cyclic if P(x,y,z)=P(y,z,x)P(x,y,z)=P(y,z,x). Prove that cyclic three variable polynomials P1,P2,P3P_1,P_2,P_3 and P4P_4 exist such that for each cyclic three variable polynomial PP, there exists a four variable polynomial QQ such that P(x,y,z)=Q(P1(x,y,z),P2(x,y,z),P3(x,y,z),P4(x,y,z))P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z)).
Solution by Mostafa Eynollahzade and Erfan Salavati
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