By Shreeram Shankar Abhyankar

This publication offers, to be used in a graduate direction or for self-study by way of graduate scholars, a well-motivated therapy of numerous issues, in particular the subsequent: algebraic remedy of numerous complicated variables; geometric method of algebraic geometry through analytic units; survey of neighborhood algebra; and survey of sheaf concept. The ebook has been written within the spirit of Weierstrass. energy sequence play the dominant function. The remedy, being algebraic, isn't really limited to complicated numbers, yet is still legitimate over any complete-valued box. This makes it acceptable to occasions bobbing up from quantity idea. while it really is really good to the complicated case, connectivity and different topological houses come to the fore. particularly, through singularities of analytic units, topological basic teams might be studied. within the transition from punctual to neighborhood, ie. from homes at some degree to houses close to some degree, the classical paintings of Osgood performs an incredible position. this offers upward push to normic types and the concept that of the Osgoodian. Following Serre, the passage from neighborhood to worldwide houses of analytic areas is facilitated by way of introducing sheaf conception. the following the elemental effects are the coherence theorems of Oka and Cartan. they're via conception normalization as a result of Oka and Zariski within the analytic and algebraic circumstances, respectively.