An algebro-geometric method for determining the rational solvability of autonomous algebraic ordinary differential equations is extended from single equations of order 1 to systems of equations of arbitrary order but dimension 1 in the algebro-geometric sense. We provide necessary conditions, for the existence of rational solutions, on the degree and on the structure at infinity of the associat...

Let K be a number field and let C be a curve of genus g > 1 defined over K. In this dissertation we describe techniques for bounding the number of K-rational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reducti...

In this paper we make use of the power series facility of Scratchpad2 , a symbolic computer language under development at the IBM Research Mathematics Department , to construct algebraic ECC on the elliptic Riemann surface of genus one. The construction method described uses products of elliptic theta series with rational characteristics and finite field coefficients to define the projective co...

We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory this class of was studied by de Jong-van Straten in [dJvS98]; they associated a germ singular plane curve to each singularity described via deformations curve. consider links singularities, equipped their canonical contact structures, develop symplectic analog Strate...