Abstract:
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.
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