Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.6 National olympiad Prove it

Inference 1 The necessary and sufficient condition for the symmetric cubic inequality satisfying $P_{3}(1,1,1)=0$
$P_{3}(x, y, z)=\lambda_{0,1} \sigma_{1}^{3}+\lambda_{0,2} \sigma_{1} \sigma_{2}+\lambda_{1,1} \sigma_{3} \geq 0$

is
$P_{3}(1,0,0)=\lambda_{0.1} \geq 0, \quad P_{3}(1,1,0)=2\left(4 \lambda_{0,1}+\lambda_{0,2}\right) \geq 0 \text {. }$

Solution

Proof: The necessity is obviously true. Now we prove the sufficiency.
Since $P_{3}(1,1,1)=0$ , according to the aforementioned method Schp, we can obtain
$P_{3}(x, y, z)=\alpha_{0,1} f_{0.1}^{(3)}+\alpha_{0,2} f_{0,2}^{(3)}=\lambda_{0.1} f_{0.1}^{(3)}+\left(4 \lambda_{0,1}+\lambda_{0,2}\right) f_{0.2}^{(3)},$

Therefore, if equation (3.1.2) holds, and by Theorem 2 we know $f_{0,1}^{(3)} \geq 0, f_{0,2}^{(3)} \geq 0$ , then equation (3.1.2) must hold. Q.E.D.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.