The commuting derivations conjecture

Abstract

This paper proves the Commuting Derivations Conjecture in dimension three: if D1 and D2 are two locally nilpotent derivations which are linearly independent and satisfy [D 1 , D 2 ] = 0 then the intersection of the kernels,is a coordinate for every zero a of p(X). Next to that, it is shown that if the Commuting Derivations Conjecture in dimension n, and the Cancellation Problem and Abhyankar-Sataye Conjecture in dimension n-1, all have an affirmative answer, then we can similarly describe all coordinates of the form p(X)Y + q(X, Z1, . . . , Zn−1). Also, conjectures about possible generalisations of the concept of "coordinate" for elements of general rings are made.

Unipotent group actions on affine varieties

“…In [7] it is conjectured that this result is true also in higher dimensions, namely, CDC(n) Commuting Derivations Conjecture. The common kernel of n commuting linearly independent locally nilpotent derivations of k [n+1] is generated by a coordinate.…”

“…This paper is originally motivated by the following result of [7]: Theorem 4. Let A = k[x, y, z] and D 1 , D 2 be two commuting locally nilpotent derivations on A which are linearly independent over A.…”

UFDs with commuting linearly independent locally nilpotent derivations

“…In the case of an affine ring A such that A = K , the fact that A X 1 ,...,X n−1 = K[c] is true, see [11]. However, the fact that A = K[c, s 1 , . .…”

“…CD(n) is recent, even though CD(2) is nothing but the well-known Rentschler's theorem [14]. To our knowledge, this Conjecture has been treated for the first time in [11], in the goal of studying coordinates of the form p(x 1 )z + q(x 1 , y 1 , . .…”

The special automorphism group of R[t]/(tm)[x1Essen1, Maubach2, Vénéreau3 2007Journal of Pure and Applied AlgebraSelf Cite10010View full textAdd to dashboardCite

“…A lot of work has been done on attempts to solve this conjecture, and also on the problem of classifying coordinates (see for example [1,4,6,8,11] and many others). One of these works, [6], studied hyperplanes in C[x, y, z, u] with a prescribed form.…”