ON THE LOCAL SOLUBILITY OF DIOPHANTINE SYSTEMS

Let p be a rational prime number. We refine Brauer's elementary diagon alisation argument to show that any system of r homogeneous polynomial s of degree d, with rational coefficients, possesses a non-trivial p-a dic solution provided only that the number of variables in this system exceeds (rd(2))(2d-1). This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analo gous conclusions in field extensions of Q(p), and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variab les required to guarantee local solubility.