Algorithms for residues and Lojasiewicz exponents

In this article, we investigate some problems of effectivity, related to algebraic residue theory. We show how matrix techniques based on Bezoutian formulations, enable us to derive new algorithms, as well as new bounds for the polynomials involved in these computations. More precisely, we focus on the computation of relations of algebraic dependency between n+1 polynomials in n variables and show how to deduce the residue of n polynomials in n variables. Applications for testing the properness of a polynomial map, for computing the Lojasiewicz exponent, and for inverting polynomial maps are also considered.