Residue calculus and effective Nullstellensatz

Multivariate residue calculus (in the spirit of J. Lipman) is developed fro m the computational point of view (for example with several variants of the classical Transformation Law), and used in order to make totally explicit the Bezout identity (and therefore the algebraic Nullstellensatz) in K[X-1, ..., X-n], where K is an infinite field of arbitrary characteristic. Such identities provide sharp size estimates for the denominator and the "diviso rs" in the Bezout identity when K is the quotient field of a factorial regu lar ring A equipped with a size (such as Z or F-p[tau(1), ..., tau(q)]). Th e estimates obtained by the authors in a previous work in the case A = Z ar e sharpened here, while analytic techniques are replaced with an algebraic approach.