Analytic and Algebraic Ideas: How to Profit from Their Complementarity

Abstract

To my friend Carlos on the occasion of his 60th birthday.Summary. Facing algebraic questions from the analytic point of view has been the guide line of the joint work which I have been pursuing for almost twenty years with Carlos Berenstein. Instead of giving an up-to-date state of the art , I will focus on a few key points which still remain to be clarified and indicate a list of prospective developments where the ideas that analysis suggests, combined with multidimensional residue theory, will certainly have to play a major role. I will point out the following crucial fact, namely that in the "dictionary" between the analytic and algebraic points of view in multidimensional residue theory, integral symbols happen to be the analytic substitutes for power series developments in terms of parameters. Briançon-Skoda theorem, which seems to be a corner stone between constructions inspired by algebraic ideas on one side and by analytic ideas on the other side, will be a leitmotiv in this talk. Such ideas need to be combined in the future with arithmetic aspects we missed up to now and which indeed imply some additional rigidity.