On some Results Related to Köthe's Conjecture

The Köthe conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil one-sided ideals. Although for more than 70 years significant progress has been made, it is still open in general. In this paper we survey some results related to the Köthe conjecture as well as some equivalent problems.

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.

Toward Clemens' Conjecture in Degrees between 10 and 24

1 Supported in part by the Norwegian Research Council for Science and the Humanities. It is a pleasure for this author to thank the Department of Mathematics of the University of Sofia for organizing the remarkable conference in Zlatograd during the period August 28-September 2, 1995. It is also a pleasure to thank the M.I.T. Department of Mathematics for its hospitality from January 1 to July 31, 1993, when this work was started. 2Supported in part by NSF grant 9400918-DMS.

Hilbert-Smith Conjecture for K - Quasiconformal Groups

MSC 2010: 30C60

Remarks on the Nagata Conjecture

2000 Mathematics Subject Classification: 14C20, 14E25, 14J26.

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

A Note About the Nowicki Conjecture on Weitzenböck Derivations

2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.