is called a locally nilpotent derivation (LND) if every element of
One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.
of characteristic zero, to give a locally nilpotent derivation on the integral domain
, finitely generated over the field, is equivalent to giving an action of the additive group
Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.
is called a locally nilpotent derivation (LND) if for every
The set of locally nilpotent derivations of a ring
Note that this set has no obvious structure: it is neither closed under addition (e.g. if
determines a locally nilpotent derivation
[3] A special case of Hilbert's 14th problem asks whether
By Zariski's finiteness theorem,[4] it is true if
is known to be finitely generated: notably, by the Maurer–Weitzenböck theorem,[6] it is the case for linear LND's of the polynomial algebra over a field of characteristic zero (by linear we mean homogeneous of degree zero with respect to the standard grading).
is a finitely generated algebra over a field of characteristic zero, then
can be computed using van den Essen's algorithm,[7] as follows.
is locally a polynomial algebra with a standard derivation.
is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.
of its coordinate ring corresponds to a natural action of
be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case
This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.
[15] Miyanishi's theorem — The kernel of every nontrivial LND of
is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial
Kaliman's theorem — Every fixed-point free action of
defines a nowhere vanishing vector field), admits a slice.
This results answers one of the conjectures from Kraft's list.
is called triangular with respect to this system of variables, if
A derivation is called triangulable if it is conjugate to a triangular one, or, equivalently, if it is triangular with respect to some system of variables.
, while by the result of Popov,[23] a fixed point set of a triangulable
is triangulable, then any hypersurface contained in the fixed-point set of the corresponding
[12] The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all
For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to