Propagation of singularities for gravity-capillary water waves

We obtain two results of propagation for the system of gravity-capillary water waves --- first the propagation of oscillations and decay at the spatial infinity and second a microlocal smoothing effect when the initial surface is non-trapping --- extending the results of Craig, Kappeler and Strauss, Wunsch and Nakamura to quasilinear dispersive equations. We also prove the existence of water waves with an asymptotically Euclidean surface and an asymptotically stationary velocity field. To obtain these results, we extend the paradifferential calculus to weighted Sobolev spaces and develop a semiclassical paradifferential calculus, we also define a family of wavefront sets --- the quasi-homogeneous wavefront sets which, at least in the Euclidean geometry, generalize the wavefront set of H\"{o}rmander, the scattering wavefront set of Melrose, the quadratic scattering wavefront set of Wunsch and the homogeneous wavefront set of Nakamura.Comment: 44 page

On a theorem of Ax and Katz

The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V. It was strengthened by Adolphson-Sperber in terms of Newton polytope of the support set G of V. In this paper we prove that for every generic algebraic variety over a number field supported on G the Adolphson-Sperber bound can be achieved on special fibre at p for a set of prime p of positive density in SpecZ. Moreover we show that if G has certain combinatorial conditional number nonzero then the above bound is achieved at special fiber at p for all but finitely many primes p.Comment: 11 page