  Let $F$ be a non-zero polynomial with integer coefficients in $N$ variables
of degree $M$. We prove the existence of an integral point of small height at
which $F$ does not vanish. Our basic bound depends on $N$ and $M$ only. We
separately investigate the case when $F$ is decomposable into a product of
linear forms, and provide a more sophisticated bound. We also relate this
problem to a certain extension of Siegel's Lemma as well as to Faltings'
version of it. Finally we exhibit an application of our results to a discrete
version of the Tarski plank problem.
