Residue calculus and effective Nullstellensatz

Multivariate residue calculus (in the spirit of J. Lipman) is developed from the computational point of view (for example with several variants of the classical Transformation Law), and used in order to make totally explicit the Bézout identity (and therefore the algebraic Nullstellensatz) in K[X₁, . . . , X_n], where K is an infinite field of arbitrary characteristic. Such identities provide sharp size estimates for the denominator and the 'divisors" in the Bézout identity when K is the quotient field of a factorial regular ring A equipped with a size (such as Z or F_p[τ₁, . . . , τ_q]). The estimates obtained by the authors in a previous work in the case A = Z are sharpened here, while analytic techniques are replaced with an algebraic approach.