Completeness in partial differential algebraic geometry

Complexity Classes and Completeness in Algebraic Geometry

We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting. Valiant's theory of algebraic/arithmetic complexity classes is an algebraic analo...

Numerical Algebraic Geometry and Differential Equations

In this paper we review applications of numerical algebraic geometry to differential equations. The techniques we address are direct solution, bootstrapping by filtering, and continuation and bifurcation. We review differential equations systems with multiple solutions and bifurcations.

Model Theory and Differential Algebraic Geometry

An Algebraic Model of Transitive Differential Geometry

One of the basic assumptions that is frequently made in the study of geometry (or physics) is that of homogeneity. I t is assumed that any point can be moved into any other by a transformation preserving the underlying geometrical structure (e.g., by a Euclidean motion in Euclidean geometry, an inhomogeneous Lorentz transformation in special relativity, etc.). The problem we shall treat in this...

The Role of Partial Differential Equations in Differential Geometry

In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the first case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the defining equations are transcendental. Indeed, even in the situatio...