Let $f(x, y, z)$ be a homogeneous polynomial and we want to prove $_{cyc}\Sigma f\ge0$. There are two different strat with the same name Buffalo Way for this problem.
Let $\min(x, y, z)=x$, expand $_{cyc}\Sigma f(x, x+u, x+v)$ and prove.
Assume $x\le y\le z$, expand $_{cyc}\Sigma f(x, x+u, x+u+v)$ and prove. Assume $x\ge y\ge z$ and repeat.
Question.
Which of these strat is stronger?
- If one of the strat is stronger: Is there a proof?
- If none of the strat is stronger than the other: Is there polynomials $f_1$ and $f_2$ whose positivity can only be proved using strat 1 and 2 respectively?
Here is an example that shows to you Buffalo way can't kill cyclic inequalities all $$\left ( y- z \right )^{2}\left ( y+ z \right )^{2}x^{5}+ 2\left ( y+ z \right )\left ( y^{4}+ z^{4} \right )x^{4}+ \left ( 8y^{3}z^{3}- y^{4}z^{2}+ z^{6}+ y^{6}- y^{2}z^{4} \right )x^{3}-$$ $$- 2yz\left ( y^{2}+ z^{2} \right )\left ( y+ z \right )^{3}x^{2}+ y^{2}z^{2}\left ( y^{4}+ 6y^{2}z^{2}+ z^{4} \right )x+ 4y^{4}z^{4}\left ( y+ z \right )\geq 0$$ holds true for $x, y, z> 0$.
Author: Art of Problem Solving/@xzlbq (Liu Bao Qian)
Yes, your strategy is up to the form of acceptance. For the given example, you should follow the second strategy. Actually there is no need.