We show that Artin’s conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r > 2.

We present a technique for finding a lower bound on the number of processors needed to achieve a given schedule of an algorithm represented as a dag for a Diophantine system. The application of this technique is illustrated with a tensor product computation. We then provide a time-minimal processor schedule that meets the computed processor lower bounds, including the one for tensor product. We...

We prove an ultrametric q-difference version of the MailletMalgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since degq and ordq define two valuations on C(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate t...

We give a survey of some classical and modern methods for solving Diophantine equations.