# Proving sup-norm of a specific polynomial (Kaisjer--Varopoulos, 1974)

This question regards a proof in the addendum (due to Kaisjer and Varopoulos) to "On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory," Varopoulos, 1974. The paper claims the following:

Let $$p(z_1,z_2,z_3):=z_1^2+z_2^2+z_3^2-2(z_1z_2+z_2z_3+z_3z_1)$$. Then $$\|p\|_\infty:=\sup\limits_{\lvert z_1\rvert,\lvert z_2\rvert,\lvert z_3\rvert\le1}\lvert p(z_1,z_2,z_3)\rvert=5$$.

The idea seems to be to use homogeneity and define $$f$$ in terms of the phase difference; letting $$z_3=e^{i\theta}z_1=e^{i\varphi}z_2$$, set $$f(\theta,\varphi):=\frac{p(z_1,z_2,z_3)}{z_3^2}$$ [the paper states $$f(\theta,g)$$, which is presumably a typo]. They then study the extrema of $$\lvert f\rvert$$ by looking at $$E\cup(E_1\cap E_2)$$ where $$E$$ is defined to be the vanishing set for $$f$$ (hence $$\lvert f\rvert$$ too), while $$E_1:=\left\{(\theta,\varphi):\mathrm{re}\left(\frac{\partial f}{\partial\theta}\cdot(1+e^{-i\theta}+e^{-i\varphi})\right)=0\right\}$$ and $$E_2$$ is the same except with the partial w.r.t. $$\varphi$$. It is then claimed that the sup-norm result is immediate, with the justification being that "On $$E_1\cap E_2$$ we have $$tg\theta+tg\varphi=0$$."

I am unclear about why $$E_1\cap E_2$$ should be where any maxima appear, and what this equality has to do with anything: $$t$$ and $$g$$ are variables that have heretofore yet appeared in this part of the proof [excepting the typo (?) earlier] and if they are real numbers would just divide out...

Any pointers are greatly appreciated.