Linear independence of powers for polynomials

Given $k\ge 2$ polynomials in $d\ge 1$ variables with coefficients in a field of characteristic $0$, such that no two are linearly dependent, we show that for any integer $r$ greater than $\max \{k\left(\genfrac{}{}{0px}{}{k-1}{2}\right),2\}$, the $r$-th powers of the polynomials are linearly independent.