For a system of polynomial equations over $\Qp$ we present an efficient construction of a single polynomial of quite small degree whose zero set over $\Qp$ coincides with the zero set over $\Qp$ of the original system. We also show that the polynomial has some other attractive features such as low additive and straight-line complexity.

The proof is based on a link established here between the above problem and some recent number theoretic result about zeros of $p$-adic forms.