SETS OF CONVERGENT POWER-SERIES GIVEN BY DIFFERENTIAL CONDITIONS

Given an ideal I subset of R{x(1), ..., x(m), y} and f is an element o f R{x(i), ..., x(m), y}, we study the subset {P.f\partial derivative(y )(P) is an element of I}. This idea comes from the classification of s ingularities of illuminated surfaces. We prove a (Hironaka-type) divis ion theorem for subsets of this form and give a Buchberger-like reduct ion algorithm. (C) 1997 Elsevier Science B.V.