The cokernel of the operator acting on a -module, II
A counterexample to a conjecture of Drużkowski and Rusek
Let F = X + H be a cubic homogeneous polynomial automorphism from to . Let be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.
A simple solution of Hilbert's fourteenth problem in dimension five
We give a short proof of a counterexample (due to Daigle and Freudenburg) to Hilbert's fourteenth problem in dimension five.
The solution of the Tame Generators Conjecture according to Shestakov and Umirbaev
The tame generators problem asked if every invertible polynomial map is tame, i.e. a finite composition of so-called elementary maps. Recently in [8] it was shown that the classical Nagata automorphism in dimension 3 is not tame. The proof is long and very technical. The aim of this paper is to present the main ideas of that proof.
The sixtieth anniversary of the Jacobian Conjecture: a new approach
We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.
The Jacobian Conjecture for symmetric Drużkowski mappings
Let k be an algebraically closed field of characteristic zero and a Drużkowski mapping of degree ≥ 2 with det JF = 1. We classify all such mappings whose Jacobian matrix JF is symmetric. It follows that the Jacobian Conjecture holds for these mappings.
A class of counterexamples to the Cancellation Problem for arbitrary rings
We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.
Recent progress on the Jacobian Conjecture
We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following:...
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