**Geometric Completion of Differential Systems**

** using Numeric-Symbolic Continuation **

*Greg Reid*,
*Chris Smith*, and
*Jan Verschelde*

#### Abstract:

In previous work symbolic algorithms have been developed which use
a finite number of exact differentiations and eliminations to
reduce over and under-determined systems of polynomially nonlinear
differential equations to involutive form. The output involutive
form enables the identification of consistent initial values, and
eases the application of exact or numerical integration methods.
However such differential-elimination algorithms,
which usually incorporate Groebner bases,
have poor complexity and are unsuited for application to systems
with approximate coefficients.

A new generation of differential-elimination algorithms is
proposed which uses homotopy continuation methods to perform the
differential-elimination process on such non-square systems.
Examples such as the classic index 3 Pendulum are given to
illustrate the new procedure. Our approach uses slicing by
random linear subspaces to intersect its jet components in
finitely many points. Generation of enough such generic points,
enables irreducible jet components of the differential system
to be interpolated.

**2000 Mathematics Subject Classification :**
Primary 34M15; Secondary 14Q99, 65H10, 68W30.

**keywords :**
Component of solutions,
DAE (Differential-Algebraic Equation), differential elimination,
differential index, embedding, geometric completion,
interpolation, irreducible component, irreducible decomposition,
generic point, homotopy continuation, numerical algebraic geometry,
numerical jet geometry, path following, polynomial system.