On Reducing a System of Equations to a Single Equation

For a system of polynomial equations over Q_p we present an efficient construction of a single polynomial of quite small degree whose zero set over Q_p coincides with the zero set over Q_p 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.