Transcendence type for almost all points in real $m$-dimensional space

Let $P$ be a polynomial of $m$ variables with integer coefficients, $\deg P$ the total degree of $P$, $H(P)$ the maximum absolute value of the coefficients of $P$, and $t(P)=\deg P+\ln H(P)$ the type of the polynomial $P$. It is shown that for almost all points $\overline\xi\in\mathbb R^m$ (in the sense of Lebesgue $m$-measure) there exists a constant $c=c(\overline\xi)>0$ such that the inequality $\ln\lvert P(\overline\xi)\rvert>-ct(P)^{m+1}$ holds for each polynomial $P\in\mathbb Z[x_1,\dots,x_m]$, $P\not\equiv0$.
Bibliography: 13 titles.