F. Leon Pritchard and William Y. Sit

Department of Mathematics and Computer Science,

Rutgers University, Newark, NJ 07102

Department of Mathematics, The City College of New York,

New York, NY 10031

leonp@andromeda.rutgers.edu, wyscc@cunyvm.cuny.edu

We consider systems of ordinary differential equations
which are polynomial in the unknown functions and their derivatives.
For a given system, we are concerned with computing algebraic
constraints on the initial conditions such that on the algebraic
variety determined by the constraint equations, the initial value
problem of the original system of differential equations has a unique
solution. For these systems, we introduce the concept of *essential degree*, and an algorithmic process called *prolongation*. The prolongation process may be repeated at most a
finite number of times, at the end of which the original system is
replaced by an equivalent system (that we called *complete*). The
length of the prolongation process is an invariant which we call the
*algebraic index* of the system. Using basic transformations, we
reduce our study to quasi-linear systems, for which we prove an
existence and uniqueness theorem, which identifies the algebraic
constraints on initial conditions. Both over-determined and
under-determined systems are studied. In the case of quasi-linear
systems, algorithms are developed and implemented.

IMACS ACA'98 Electronic Proceedings