COMPUTATION OF MULTIVARIABLE DIFFERENTIAL DIMENSION POLYNOMIALS

                        Alexander Levin
                    Department of Mathematics
                 The Catholic University of America
                     Washington, D.C. 20064      
                        e-mail: LEVIN@cua.edu
          
Abstract:
          Let  F  be a differential field of zero characteristic and let a
partition of the basic set of derivation operators of the field  F  into  p  
disjoint subsets be fixed. We introduce a special type of reduction in the ring
of differential polynomials over  F  and develop the appropriate technique of
characteristic sets that allows to prove the existence and find methods of
computation of differential dimension polynomials in  p  variables associated
with finitely generated differential field extensions of  F .  We also show
that the multivariable differential dimension polynomials (associated with
various partitions of the basic set of derivation operators) give differential
birational invariants that are not carried by the Kolchin dimension polynomial.