On the Local Solubility of Diophantine Systems

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Let p be a rational prime number. We refine Brauer's elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (rd2)2d−1. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Qp, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.

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