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