On the smallest integer vector at which a multivariable polynomial does not vanish

We prove that for any polynomial $P$ of degree $d$ in $C[x_1,dots,x_n]$ there exists a vector $(u_1,dots,u_n) in Z^n$ such that $P(u_1,dots,u_n) ne 0$ and $sum_{i=1}^n |u_i| leq min{d, lfloor (d+n)/2 rfloor}$. We also show that this bound is best possible. Similarly, for any $P in C[x_1,dots,x_n]$ of degree $d$ and any real number $p geq log 3/log 2$ there is a vector $(u_1,dots,u_n) in Z^n$ such that $P(u_1,dots,u_n) ne 0$ and $sum_{i=1}^n |u_i|^p leq max{1+lfloor d/2 rfloor^p, lfloor (d+1)/2 rfloor^p}$. The latter bound is also best possible for every $n geq 2$