Linear and Complex Analysis Problem Book 3: Part II by N. K. Nikolski (auth.), Victor P. Havin, Nikolai K. Nikolski

By N. K. Nikolski (auth.), Victor P. Havin, Nikolai K. Nikolski (eds.)

The 2-volume-book is an up to date, reorganized and significantly enlarged model of the former variation of the study challenge publication in research (LNM 1043), a suite conventional to many analysts, that has sparked off a lot learn. This new version, created in a joint attempt via a wide staff of analysts, is, like its predecessor, a set of unsolved difficulties of contemporary research designed as informally written mini-articles, each one containing not just an announcement of an issue but in addition ancient and metho- dological reviews, motivation, conjectures and dialogue of attainable connections, of believable ways in addition to an inventory of references. There are actually 342 of those mini- articles, virtually two times as many as within the prior variation, even though a great deal of them were solved!

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PROBLEM 2. e. for any F in A r such that, for any x E C n, the g e r m os F at x is in the integral closure o / t h e ideal Iz in 0 ~ , does one have F inf(n'm) E I (or at least F ~ E I, w i t h s o m e e x p o n e n t t~ depending only on n)? PROBLEM 3. W h e n m << n and the zero set of I is n - m dimensional, is the ideal I d o s e d in air ? A positive answer to these three questions (with ~ = 2 inf(m + 1, n)) is given in [4] in the very particular case where r a n k F -- 1. 9 is crucial). e.

Its definition requires some preparation. The set ~ := ~r-l(A) is called a ~r-fiber over A. A set is ~r-symmetric if it may be represented as a union of some collection of ~r-fibers. Let V be a ~r-symmetric set. A function f : V ---* C is called ~r-symmetric if it takes equal values on ~r-fibers ~ C V. For a fixed point )~ E C denote O(~) the set of germs of functions locally analytic in neighbourhoods of ~r-fiber ~ over ~. O . ( ~ ) is a subring of O(~) consisting of 7r-germs of ~r-symmetric functions locally analytic in r - s y m m e t r i c neighbourhoods of ~.

We list a number of specific questions along these lines. (a) Suppose f, g, h E EP(#) and f = gh. If g and h are weakly invertible, is f weakly invertible? Conversely, if f is weakly invertible, are g and h weakly invertible? (b) If f 9 E ' ( # ) is nonvanishing and ~ 9 Eq(tt) for some q, q > O, is f weakly invertible ? (c) If f 9 EP(#) is weakly invertible and a > O, is f~ weakly invertible in EP/~(#)? (d) If f 9 E ' ( # ) and f is weakly invertible in Eq(#) for some q, q < p, is f weakly invertible in EP(#) ?

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