w_2} := -1/2- (\sqrt{3}/2 )\i\,= \, {\bf w_1}^2.$ We have $$\begin{aligned}
r(\bar{x}) = 
\frac{1}{3} \left[ (x_0\,+\,x_1\,+\,x_2 )^k \,+  \, 
(x_0 \, + \,{ \bf w_1} x_1 + {\bf  w_2} x_2)^k \,+ \, (x_0 \,+ \,{ \bf w_2} x_1 \,+ \, 
{\bf w_1}  x_2)^k \right] \label{RCOM}\end{aligned}$$ The preceding equation is well known and easy to prove from basic properties of roots of unity. Note that in derivatives $\frac{d}{dx_i} r(\bar{x})$ the roots of unity are treated as constants.

For $ x_i, y_i >0$ we define (convention $\alpha^\alpha=1$ for $\alpha=\omega_i$ or $\alpha=\lambda_i$ and $\alpha =0)$ $$\begin{aligned}
\Psi( \bar{\omega}\, ,\, \bar{\lambda}\, ,\, \bar{x} \,,\, \bar{y}  \, )\,= \, 
\prod_{i=0,1,2}\left( \frac{q(x_i )}{\omega_i q(s)} \right)^{\omega_i} \cdot
\left[ \prod_{i=0, 1, 2} \left(\frac{\lambda_i s }{x_i y_i}\right)^{\lambda_i } \right]^{k \gamma}
 r(y_0, y_1, y_2)^\gamma  \nonumber \end{aligned}$$ With $\omega_i=\lambda_i=1/3, a_i=s(k, \gamma)=s, $ and $c_i=1,$ we have $\Psi(\overline{\omega}, \overline{\lambda}, \bar{a}, \bar{c})=
3\cdot (1/3)^{k\gamma}\cdot ((1/3)3^k)^\gamma= 3^{1-\gamma} $ (use (\[RCOM\]).)

\[LEMBA\] $ N(\bar{w}, \bar{l})/N_0 \, \, < \, \, \Psi( \bar{\omega}\, ,\, \bar{\lambda}\, ,\, \bar{a} \,,\, \bar{c}  \, )^n\cdot O(n)^{3/2}$ for any $\bar{a}, \bar{c},  a_i, c_i>0.$

${n \choose \bar{w}} \le \prod_i (1/\omega_i)^{\omega_i n}$ for all $\bar{w}, $ ([@MITZEN], page 228 ) $\prod_{i=0, 1, 2} M(l_i, w_i)/N_0 \, \le \, \\
 \prod_i \left( (l_i/(a_ie))^{l_i} q(a_i)^{w_i} 