There is given a prime number $p$ and two positive integers, $k$ and $n$. Determine the smallest nonnegative integer $d$ for which there exists a polynomial $f\left(x_1\mathrm{,}\text{...}\mathrm{,}{x}_n\right)$ on $n$ variables, with degree $d$ and having integer coefficients that satisfies the following property: for arbitrary $a_1\mathrm{,}\text{...}\mathrm{,}{a}_n\in \left\{\mathrm{0,1}\right\}$, $p$ divides $f\left(a_1\mathrm{,}\text{...}\mathrm{,}{a}_n\right)$ if and only if $p^k$ divides $a_1+\text{...}+a_n$.