Derivations and radicals of polynomial ideals over fields of arbitrary characteristic

The purpose of this paper is to give a complete effective solution to the problem of computing radicals of polynomial ideals over general fields of arbitrary characteristic, We prove that Seidenberg's "Condition P" is both a necessary and sufficient property of the coefficient field in order to be able to perform this computation, Since Condition P is an expensive additional requirement on the ground field, we use derivations and ideal quotients to recover as much of the radical as possible, If we have a basis for the vector space of derivations on our ground field, then the problem of computing radicals can be reduced to computing pth roots of elements in finite dimensional algebras, (C) 2002 Elsevier Science Ltd. All rights reserved.