Order function for almost all numbers

For almost all pointsxgrexist $\xi\in R^m$ ($m>2$) the inequality
$$ \sup\ln\frac1{|P(\xi)|}\ll(\ln u)^{m+2}, $$
is valid, where the upper bound is taken over all nonzero polynomials $P$ for which $\exp(\operatorname{deg}P)L(P)<u$ where $L(P)$ is the sum of the moduli of the coefficients of $P$.
When $m=1$ the exponent of the right side is equal to 2.