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sum of square like a discriminent

Source: 2017 China TST 5 P5

April 14, 2017
algebrapolynomialChina TST

Problem Statement

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that B(x,y)2−4A(x,y)C(x,y)=−R(x,y)2B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that F(x,y,z)2+G(x,y,z)2=A(x,y)z2+B(x,y)z+C(x,y)F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y) if for any x, y, z real numbers A(x,y)z2+B(x,y)z+C(x,y)≥0A(x,y)z^2+B(x,y)z+C(x,y)\ge 0
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