Locally nilpotent derivation

Let $A$ be a ring. Recall that a derivation of $A$ is a map $\backslash partial\backslash colon\backslash ,\; A\backslash to\; A$ satisfying the Leibniz rule $\backslash partial\; (ab)=(\backslash partial\; a)b+a(\backslash partial\; b)$ for any $a,b\backslash in\; A$. If $A$ is an algebra over a field $k$, we additionally require $\backslash partial$ to be $k$-linear, so $k\backslash subseteq\; \backslash ker\; \backslash partial$.

A derivation $\backslash partial$ is called a locally nilpotent derivation (LND) if for every $a\; \backslash in\; A$, there exists a positive integer $n$ such that $\backslash partial^\{n\}(a)=0$.

If $A$ is graded, we say that a locally nilpotent derivation $\backslash partial$ is homogeneous (of degree $d$) if $\backslash deg\; \backslash partial\; a=\backslash deg\; a\; +d$ for every $a\backslash in\; A$.

The set of locally nilpotent derivations of a ring $A$ is denoted by $\backslash operatorname\{LND\}(A)$. Note that this set has no obvious structure: it is neither closed under addition (e.g. if $\backslash partial\_\{1\}=y\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\}$, $\backslash partial\_\{2\}=x\backslash tfrac\{\backslash partial\}\{\backslash partial\; y\}$ then $\backslash partial\_\{1\},\backslash partial\_\{2\}\backslash in\; \backslash operatorname\{LND\}(k[x,y])$ but $(\backslash partial\_\{1\}+\backslash partial\_\{2\})^\{2\}(x)=x$, so $\backslash partial\_\{1\}+\backslash partial\_\{2\}\backslash not\backslash in\; \backslash operatorname\{LND\}(k[x,y])$) nor under multiplication by elements of $A$ (e.g. $\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\}\backslash in\; \backslash operatorname\{LND\}(k[x])$, but $x\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\}\backslash not\backslash in\backslash operatorname\{LND\}(k[x])$). However, if $[\backslash partial\_\{1\},\backslash partial\_\{2\}]=0$ then $\backslash partial\_\{1\},\backslash partial\_\{2\}\backslash in\; \backslash operatorname\{LND\}(A)$ implies $\backslash partial\_\{1\}+\backslash partial\_\{2\}\backslash in\; \backslash operatorname\{LND\}(A)$ and if $\backslash partial\backslash in\; \backslash operatorname\{LND\}(A)$, $h\backslash in\backslash ker\backslash partial$ then $h\backslash partial\backslash in\; \backslash operatorname\{LND\}(A)$.

Let $A$ be an algebra over a field $k$ of characteristic zero (e.g. $k=\backslash mathbb\{C\}$). Then there is a one-to-one correspondence between the locally nilpotent $k$-derivations on $A$ and the actions of the additive group $\backslash mathbb\{G\}\_\{a\}$ of $k$ on the affine variety $\backslash operatorname\{Spec\}\; A$, as follows.{{cite book|last1=Freudenburg|first1=G.|title=Algebraic theory of locally nilpotent derivations|date=2006|publisher=Springer-Verlag|location=Berlin| isbn=978-3-540-29521-1| citeseerx=10.1.1.470.10}}

A $\backslash mathbb\{G\}\_\{a\}$-action on $\backslash operatorname\{Spec\}\; A$ corresponds to a $k$-algebra homomorphism $\backslash rho\backslash colon\; A\backslash to\; A[t]$. Any such $\backslash rho$ determines a locally nilpotent derivation $\backslash partial$ of $A$ by taking its derivative at zero, namely $\backslash partial=\backslash epsilon\; \backslash circ\; \backslash tfrac\{d\}\{dt\}\backslash circ\; \backslash rho,$ where $\backslash epsilon$ denotes the evaluation at $t=0$.

Conversely, any locally nilpotent derivation $\backslash partial$ determines a homomorphism $\backslash rho\backslash colon\; A\backslash to\; A[t]$ by $\backslash rho\; =\; \backslash exp\; (t\backslash partial)=\backslash sum\_\{n=0\}^\{\backslash infty\}\backslash frac\{t^\{n\}\}\{n!\}\backslash partial^\{n\}.$

It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if $\backslash alpha\backslash in\; \backslash operatorname\{Aut\}\; A$ and $\backslash partial\backslash in\; \backslash operatorname\{LND\}(A)$ then $\backslash alpha\backslash circ\backslash partial\; \backslash circ\; \backslash alpha^\{-1\}\backslash in\; \backslash operatorname\{LND\}(A)$ and $\backslash exp(t\backslash cdot\; \backslash alpha\backslash circ\backslash partial\; \backslash circ\; \backslash alpha^\{-1\})=\backslash alpha\; \backslash circ\; \backslash exp(t\backslash partial)\backslash circ\; \backslash alpha^\{-1\}$

The algebra $\backslash ker\; \backslash partial$ consists of the invariants of the corresponding $\backslash mathbb\{G\}\_\{a\}$-action. It is algebraically and factorially closed in $A$. A special case of Hilbert's 14th problem asks whether $\backslash ker\; \backslash partial$ is finitely generated, or, if $A=k[X]$, whether the quotient $X/\backslash !/\backslash mathbb\{G\}\_\{a\}$ is affine. By Zariski's finiteness theorem,{{cite journal|last1=Zariski|first1=O.|title=Interprétations algébrico-géométriques du quatorzième problème de Hilbert|journal=Bull. Sci. Math. (2)|date=1954|volume=78|pages=155–168}} it is true if $\backslash dim\; X\backslash leq\; 3$. On the other hand, this question is highly nontrivial even for $X=\backslash mathbb\{C\}^\{n\}$, $n\backslash geq\; 4$. For $n\backslash geq\; 5$ the answer, in general, is negative.{{cite journal|last1=Derksen|first1=H. G. J.|title=The kernel of a derivation|journal=J. Pure Appl. Algebra|date=1993|volume=84|issue=1|pages=13–16|doi=10.1016/0022-4049(93)90159-Q|doi-access=free}} The case $n=4$ is open.

However, in practice it often happens that $\backslash ker\backslash partial$ is known to be finitely generated: notably, by the Maurer–Weitzenböck theorem,{{cite journal|last1=Seshadri|first1=C.S.|title=On a theorem of Weitzenböck in invariant theory|journal=J. Math. Kyoto Univ.|date=1962|volume=1|issue=3|pages=403–409|doi=10.1215/kjm/1250525012|url=https://projecteuclid.org/download/pdf_1/euclid.kjm/1250525012|doi-access=free}} 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).

Assume $\backslash ker\; \backslash partial$ is finitely generated. If $A=k[g\_1,\backslash dots,\; g\_n]$ is a finitely generated algebra over a field of characteristic zero, then $\backslash ker\backslash partial$ can be computed using van den Essen's algorithm,{{cite book|last1=van den Essen|first1=A.|title=Polynomial automorphisms and the Jacobian conjecture|date=2000|publisher=Birkhäuser Verlag|location=Basel|isbn=978-3-7643-6350-5|doi=10.1007/978-3-0348-8440-2|s2cid=252433637 }} as follows. Choose a local slice, i.e. an element $r\backslash in\; \backslash ker\; \backslash partial^\{2\}\backslash setminus\; \backslash ker\; \backslash partial$ and put $f=\backslash partial\; r\backslash in\; \backslash ker\backslash partial$. Let $\backslash pi\_\{r\}\backslash colon\backslash ,\; A\backslash to\; (\backslash ker\; \backslash partial)\_\{f\}$ be the Dixmier map given by $\backslash pi\_\{r\}(a)=\backslash sum\_\{n=0\}^\{\backslash infty\}\backslash frac\{(-1)^\{n\}\}\{n!\}\backslash partial^\{n\}(a)\backslash frac\{r^\{n\}\}\{f^\{n\}\}$. Now for every $i=1,\backslash dots,\; n$, chose a minimal integer $m\_\{i\}$ such that $h\_\{i\}\backslash colon\; =\; f^\{m\_\{i\}\}\backslash pi\_\{r\}(g\_\{i\})\backslash in\; \backslash ker\backslash partial$, put $B\_\{0\}=k[h\_\{1\},\backslash dots,\; h\_\{n\},f]\backslash subseteq\; \backslash ker\; \backslash partial$, and define inductively $B\_\{i\}$ to be the subring of $A$ generated by $\backslash \{h\backslash in\; A:\; fh\backslash in\; B\_\{i-1\}\backslash \}$. By induction, one proves that $B\_\{0\}\backslash subset\; B\_\{1\}\backslash subset\; \backslash dots\; \backslash subset\backslash ker\; \backslash partial$ are finitely generated and if $B\_\{i\}=B\_\{i+1\}$ then $B\_\{i\}=\backslash ker\; \backslash partial$, so $B\_\{N\}=\backslash ker\; \backslash partial$ for some $N$. Finding the generators of each $B\_\{i\}$ and checking whether $B\_\{i\}=B\_\{i+1\}$ is a standard computation using Gröbner bases.

Assume that $\backslash partial\backslash in\backslash operatorname\{LND\}(A)$ admits a slice, i.e. $s\backslash in\; A$ such that $\backslash partial\; s=1$. The slice theorem asserts that $A$ is a polynomial algebra $(\backslash ker\backslash partial)\; [s]$ and $\backslash partial=\backslash tfrac\{d\}\{ds\}$.

For any local slice $r\backslash in\backslash ker\backslash partial\; \backslash setminus\; \backslash ker\backslash partial^\{2\}$ we can apply the slice theorem to the localization $A\_\{\backslash partial\; r\}$, and thus obtain that $A$ is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient $\backslash pi\backslash colon\backslash ,X\backslash to\; X//\backslash mathbb\{G\}\_\{a\}$ is affine (e.g. when $\backslash dim\; X\backslash leq\; 3$ by the Zariski theorem), then it has a Zariski-open subset $U$ such that $\backslash pi^\{-1\}(U)$ is isomorphic over $U$ to $U\backslash times\; \backslash mathbb\{A\}^\{1\}$, where $\backslash mathbb\{G\}\_\{a\}$ acts by translation on the second factor.

However, in general it is not true that $X\backslash to\; X//\backslash mathbb\{G\}\_\{a\}$ is locally trivial. For example,{{cite journal|last1=Deveney|first1=J.|last2=Finston|first2=D.|title=A proper $\backslash mathbb\{G\}\_\{a\}$-action on $\backslash mathbb\{C\}^\{5\}$ which is not locally trivial|journal=Proc. Amer. Math. Soc.|date=1995|volume=123|issue=3|pages=651–655| doi=10.1090/S0002-9939-1995-1273487-0 |doi-access=free|jstor=2160782}} let $\backslash partial=u\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\}+v\backslash tfrac\{\backslash partial\}\{\backslash partial\; y\}+(1+uy^2)\backslash tfrac\{\backslash partial\}\{\backslash partial\; z\}\backslash in\; \backslash operatorname\{LND\}(\backslash mathbb\{C\}[x,y,z,u,v])$. Then $\backslash ker\backslash partial$ is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.

If $\backslash dim\; X=3$ then $\backslash Gamma=X\backslash setminus\; U$ is a curve. To describe the $\backslash mathbb\{G\}\_\{a\}$-action, it is important to understand the geometry $\backslash Gamma$. Assume further that $k=\backslash mathbb\{C\}$ and that $X$ is smooth and contractible (in which case $S$ is smooth and contractible as well{{cite journal|last1=Kaliman|first1=S|last2=Saveliev|first2=N.|title=$\backslash mathbb\{C\}\_\{+\}$-Actions on contractible threefolds|journal=Michigan Math. J.|date=2004|volume=52|issue=3|pages=619–625|doi=10.1307/mmj/1100623416|url=https://projecteuclid.org/download/pdf_1/euclid.mmj/1100623416|arxiv=math/0209306|s2cid=15020160}}) and choose $\backslash Gamma$ to be minimal (with respect to inclusion). Then Kaliman proved that each irreducible component of $\backslash Gamma$ is a polynomial curve, i.e. its normalization is isomorphic to $\backslash mathbb\{C\}^\{1\}$. The curve $\backslash Gamma$ for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in $\backslash mathbb\{C\}^\{2\}$, so $\backslash Gamma$ may not be irreducible. However, it is conjectured that $\backslash Gamma$ is always contractible.{{cite book| last1=Kaliman|first1=S.| contribution=Actions of $\backslash mathbb\{C\}^\{*\}$ and $\backslash mathbb\{C\}\_\{+\}$ on affine algebraic varieties| date=2009|title=Algebraic geometry-Seattle 2005. Part 2|volume=80| pages=629–654|doi=10.1090/pspum/080.2/2483949|contribution-url=http://www.math.miami.edu/~kaliman/library/seattle.paper.pdf|series=Proceedings of Symposia in Pure Mathematics|isbn=9780821847039}}

The standard coordinate derivations $\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_i\}$ of a polynomial algebra $k[x\_1,\backslash dots,\; x\_n]$ are locally nilpotent. The corresponding $\backslash mathbb\{G\}\_a$-actions are translations: $t\backslash cdot\; x\_\{i\}=x\_\{i\}+t$, $t\backslash cdot\; x\_\{j\}=x\_\{j\}$ for $j\backslash neq\; i$.

Source:{{cite journal|last1=Freudenburg|first1=G.|title=Actions of $\backslash mathbb\{G\}\_a$ on $\backslash mathbb\{A\}^\{3\}$ defined by homogeneous derivations|journal=Journal of Pure and Applied Algebra|date=1998|volume=126|issue=1|pages=169–181|doi=10.1016/S0022-4049(96)00143-0|doi-access=free}}

Let $f\_1=x\_1x\_3-x\_2^2$, $f\_2=x\_3f\_1^2+2x\_1^2x\_2f\_1+x^5$, and let $\backslash partial$ be the Jacobian derivation $\backslash partial(f\_\{3\})=\backslash det\; \backslash left[\backslash tfrac\{\backslash partial\; f\_\{i\}\}\{\backslash partial\; x\_\{j\}\}\backslash right]\_\{i,j=1,2,3\}$. Then $\backslash partial\backslash in\; \backslash operatorname\{LND\}(k[x\_1,x\_2,x\_3])$ and $\backslash operatorname\{rank\}\backslash partial=3$ (see below); that is, $\backslash partial$ annihilates no variable. The fixed point set of the corresponding $\backslash mathbb\{G\}\_\{a\}$-action equals $\backslash \{x\_1=x\_2=0\backslash \}$.

Consider $Sl\_2(k)=\backslash \{ad-bc=1\backslash \}\backslash subseteq\; k^\{4\}$. The locally nilpotent derivation $a\backslash tfrac\{\backslash partial\}\{\backslash partial\; b\}+c\backslash tfrac\{\backslash partial\}\{\backslash partial\; d\}$ of its coordinate ring corresponds to a natural action of $\backslash mathbb\{G\}\_a$ on $Sl\_2(k)$ via right multiplication of upper triangular matrices. This action gives a nontrivial $\backslash mathbb\{G\}\_a$-bundle over $\backslash mathbb\{A\}^\{2\}\backslash setminus\; \backslash \{(0,0)\backslash \}$. However, if $k=\backslash mathbb\{C\}$ then this bundle is trivial in the smooth category{{cite journal|last1=Dubouloz|first1=A.|last2=Finston|first2=D.|title=On exotic affine 3-spheres|journal=J. Algebraic Geom.|date=2014|volume=23| issue=3|pages=445–469| doi=10.1090/S1056-3911-2014-00612-3| arxiv=1106.2900|s2cid=119651964 }}

Let $k$ be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case $k=\backslash mathbb\{C\}${{cite journal|last1=Daigle|first1=D.|last2=Kaliman|first2=S.|title=A note on locally nilpotent derivations and variables of $k[X,Y,Z]$|journal=Canad. Math. Bull.|date=2009|volume=52|issue=4| pages=535–543|doi=10.4153/CMB-2009-054-5|doi-access=free|url=http://www.math.miami.edu/~kaliman/library/canada.daigle-kaliman.pdf}}) and let $A=k[x\_1,\backslash dots,\; x\_n]$ be a polynomial algebra.

{{math theorem | name = Rentschler's theorem | math_statement =

Every LND of $k[x\_1,x\_2]$ can be conjugated to $f(x\_1)\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_2\}$ for some $f(x\_1)\backslash in\; k[x\_1]$. This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.{{cite journal|last1=Rentschler|first1=R.|title=Opérations du groupe additif sur le plan affine|journal=Comptes Rendus de l'Académie des Sciences, Série A-B|date=1968|volume=267|pages=A384–A387}}}}

{{math theorem | name = Miyanishi's theorem | math_statement =

The kernel of every nontrivial LND of $A=k[x\_1,x\_2,x\_3]$ is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial $\backslash mathbb\{G\}\_\{a\}$-action on $\backslash mathbb\{A\}^\{3\}$ is isomorphic to $\backslash mathbb\{A\}^\{2\}$.{{cite book|last1=Miyanishi|first1=M.|contribution=Normal affine subalgebras of a polynomial ring|title=Algebraic and Topological Theories (Kinosaki, 1984)|date=1986|pages=37–51|url=https://www.researchgate.net/publication/41754985}}{{cite book|last1=Sugie|first1=T.|chapter=Algebraic Characterization of the Affine Plane and the Affine 3-Space |title=Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988)|volume=80|date=1989| pages=177–190|doi=10.1007/978-1-4612-3702-0_12|isbn=978-1-4612-8219-8|publisher=Birkhäuser Boston|series=Progress in Mathematics}}

In other words, for every $0\backslash neq\; \backslash partial\; \backslash in\; \backslash operatorname\{LND\}(A)$ there exist $f\_\{1\},f\_\{2\}\backslash in\; A$ such that $\backslash ker\backslash partial=k[f\_\{1\},f\_\{2\}]$ (but, in contrast to the case $n=2$, $A$ is not necessarily a polynomial ring over $\backslash ker\; \backslash partial$). In this case, $\backslash partial$ is a Jacobian derivation: $\backslash partial(f\_\{3\})\; =\; \backslash det\backslash left[\backslash tfrac\{\backslash partial\; f\_\{i\}\}\{\backslash partial\; x\_\{j\}\}\backslash right]\_\{i,j=1,2,3\}$.{{cite journal|last1=D.|first1=Daigle|title=On kernels of homogeneous locally nilpotent derivations of $k[X,Y,Z]$|journal=Osaka J. Math.|date=2000|volume=37|issue=3|pages=689–699|url=https://projecteuclid.org/download/pdf_1/euclid.ojm/1200789363}}}}

{{math theorem | name = Zurkowski's theorem | math_statement =

Assume that $n=3$ and $\backslash partial\backslash in\; \backslash operatorname\{LND\}(A)$ is homogeneous relative to some positive grading of $A$ such that $x\_1,x\_2,x\_3$ are homogeneous. Then $\backslash ker\backslash partial=k[f,g]$ for some homogeneous $f,g$. Moreover, if $\backslash deg\; x\_\{1\},\backslash deg\; x\_\{2\},\backslash deg\; x\_\{3\}$ are relatively prime, then $\backslash deg\; f,\backslash deg\; g$ are relatively prime as well.{{cite web|last1=Zurkowski|first1=V.D.|title=Locally finite derivations|url=http://www.math.ru.nl/~maubach/Research/zurkowski.pdf}}}}

{{math theorem | name = Bonnet's theorem | math_statement =

A quotient morphism $\backslash mathbb\{A\}^\{3\}\backslash to\; \backslash mathbb\{A\}^\{2\}$ of a $\backslash mathbb\{G\}\_\{a\}$-action is surjective. In other words, for every $0\backslash neq\; \backslash partial\; \backslash in\; \backslash operatorname\{LND\}(A)$, the embedding $\backslash ker\backslash partial\backslash subseteq\; A$ induces a surjective morphism $\backslash operatorname\{Spec\}A\backslash to\; \backslash operatorname\{Spec\}\backslash ker\backslash partial$.{{cite journal|last1=Bonnet|first1=P.|title=Surjectivity of quotient maps for algebraic $(\backslash mathbb\{C\},+)$-actions and polynomial maps with contractible fibers|journal=Transform. Groups|date=2002|volume=7|issue=1|pages=3–14|doi=10.1007/s00031-002-0001-6|arxiv=math/0602227}}

This is no longer true for $n\backslash geqslant\; 4$, e.g. the image of a quotient map $\backslash mathbb\{A\}^\{4\}\backslash to\backslash mathbb\{A\}^\{3\}$ by a $\backslash mathbb\{G\}\_\{a\}$-action $t\backslash cdot\; (x\_1,x\_2,x\_3,x\_4)=(x\_1,x\_2,x\_3-tx\_2,x\_4+tx\_1)$ (which corresponds to a LND given by $x\_1\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_4\}-x\_2\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_3\})$ equals $\backslash mathbb\{A\}^\{3\}\backslash setminus\; \backslash \{(x\_1,x\_2,x\_3):\; x\_\{1\}=x\_\{2\}=0,x\_\{3\}\backslash neq\; 0\backslash \}$.}}

{{math theorem | name = Kaliman's theorem | math_statement =

Every fixed-point free action of $\backslash mathbb\{G\}\_\{a\}$ on $\backslash mathbb\{A\}^\{3\}$ is conjugate to a translation. In other words, every $\backslash partial\; \backslash in\; \backslash operatorname\{LND\}(A)$ such that the image of $\backslash partial$ generates the unit ideal (or, equivalently, $\backslash partial$ defines a nowhere vanishing vector field), admits a slice. This results answers one of the conjectures from Kraft's list.{{cite journal|last1=Kaliman|first1=S.|title=Free $\backslash mathbb\{C\}\_\{+\}$-actions on $\backslash mathbb\{C\}^3$ are translations|journal=Invent. Math.|date=2004|volume=156|issue=1|pages=163–173|doi=10.1007/s00222-003-0336-1|url=http://www.math.miami.edu/~kaliman/library/Ka.invent.2004.c+.pdf|arxiv=math/0207156|s2cid=15769378 }}

Again, this result is not true for $n\backslash geqslant\; 4$:{{cite journal|last1=Winkelmann|first1=J.|title=On free holomorphic $\backslash mathbb\{C\}$-actions on $\backslash mathbb\{C\}^n$ and homogeneous Stein manifolds|journal=Math. Ann.|date=1990|volume=286|issue=1–3|pages=593–612|doi=10.1007/BF01453590|doi-access=free|url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN235181684_0286/PPN235181684_0286___LOG_0038.pdf}} e.g. consider the $x\_1\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_2\}+\; x\_2\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_3\}+(x\_2^2-2x\_1\; x\_3-1)\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_\{4\}\}\backslash in\; \backslash operatorname\{LND\}(\backslash mathbb\{C\}[x\_\{1\},x\_\{2\},x\_\{3\},x\_\{4\}])$. The points $(x\_1,1,0,0)$ and $(x\_1,-1,0,0)$ are in the same orbit of the corresponding $\backslash mathbb\{G\}\_a$-action if and only if $x\_\{1\}\backslash neq\; 0$; hence the (topological) quotient is not even Hausdorff, let alone homeomorphic to $\backslash mathbb\{C\}^\{3\}$.}}

{{math theorem | name = Principal ideal theorem | math_statement =

Let $\backslash partial\backslash in\backslash operatorname\{LND\}(A)$. Then $A$ is faithfully flat over $\backslash ker\backslash partial$. Moreover, the ideal $\backslash ker\; \backslash partial\; \backslash cap\; \backslash operatorname\{im\}\backslash partial$ is principal in $A$.}}

Let $f\_1,\backslash dots,f\_n$ be any system of variables of $A$; that is, $A=k[f\_1,\backslash dots,\; f\_n]$. A derivation of $A$ is called triangular with respect to this system of variables, if $\backslash partial\; f\_1\backslash in\; k$ and $\backslash partial\; f\_\{i\}\; \backslash in\; k[f\_1,\backslash dots,f\_\{i-1\}]$ for $i=2,\backslash dots,n$. 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. Every triangular derivation is locally nilpotent. The converse is true for $\backslash leq\; 2$ by Rentschler's theorem above, but it is not true for $n\backslash geq\; 3$.

;Bass's example

The derivation of $k[x\_1,x\_2,x\_3]$ given by $x\_1\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_2\}+2x\_2x\_1\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_3\}$ is not triangulable.{{cite journal|last1=Bass|first1=H.|title=A non-triangular action of $\backslash mathbb\{G\}\_\{a\}$ on $\backslash mathbb\{A\}^\{3\}$|journal=Journal of Pure and Applied Algebra|date=1984|volume=33|issue=1|pages=1–5|doi=10.1016/0022-4049(84)90019-7|doi-access=free}} Indeed, the fixed-point set of the corresponding $\backslash mathbb\{G\}\_\{a\}$-action is a quadric cone $x\_2x\_3=x\_2^2$, while by the result of Popov,{{cite book|last1=Popov|first1=V. L.|title=Algebraic Groups Utrecht 1986 |chapter=On actions of $\backslash mathbb\{G\}\_a$ on $\backslash mathbb\{A\}^n$ |volume=1271|pages=237–242|doi=10.1007/BFb0079241|series=Lecture Notes in Mathematics|year=1987|isbn=978-3-540-18234-4}} a fixed point set of a triangulable $\backslash mathbb\{G\}\_\{a\}$-action is isomorphic to $Z\backslash times\; \backslash mathbb\{A\}^\{1\}$ for some affine variety $Z$; and thus cannot have an isolated singularity.

{{math theorem | name = Freudenburg's theorem | math_statement =

The above necessary geometrical condition was later generalized by Freudenburg.{{cite journal|last1=Freudenburg|first1=G.|title=Triangulability criteria for additive group actions on affine space|journal=J. Pure Appl. Algebra|date=1995|volume=105|issue=3|pages=267–275|doi=10.1016/0022-4049(96)87756-5|doi-access=free}} To state his result, we need the following definition:

A corank of $\backslash partial\backslash in\; \backslash operatorname\{LND\}(A)$ is a maximal number $j$ such that there exists a system of variables $f\_1,\backslash dots,\; f\_n$ such that $f\_1,\backslash dots,\; f\_j\backslash in\backslash ker\backslash partial$. Define $\backslash operatorname\{rank\}\backslash partial$ as $n$ minus the corank of $\backslash partial$.

We have $1\backslash leq\; \backslash operatorname\{rank\}\backslash partial\; \backslash leq\; n$ and $\backslash operatorname\{rank\}(\backslash partial)=1$ if and only if in some coordinates, $\backslash partial=h\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_\{n\}\}$ for some $h\backslash in\; k[x\_1,\backslash dots,x\_\{n-1\}]$.

Theorem: If $\backslash partial\backslash in\; \backslash operatorname\{LND\}(A)$ is triangulable, then any hypersurface contained in the fixed-point set of the corresponding $\backslash mathbb\{G\}\_\{a\}$-action is isomorphic to $Z\backslash times\; \backslash mathbb\{A\}^\{\backslash operatorname\{rank\}\; \backslash partial\}$.

In particular, LND's of maximal rank $n$ cannot be triangulable. Such derivations do exist for $n\backslash geq\; 3$: the first example is the (2,5)-homogeneous derivation (see above), and it can be easily generalized to any $n\backslash geq\; 3$.}}

{{Main|Makar-Limanov invariant}}

The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all $\backslash mathbb\{G\}\_\{a\}$-actions, is called "Makar-Limanov invariant" and is an important algebraic invariant of an affine variety. For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to $\backslash mathbb\{C\}^\{3\}$, it is not.{{cite journal|last1=Kaliman|first1=S.|last2=Makar-Limanov|first2=L.|title=On the Russell-Koras contractible threefolds|journal=J. Algebraic Geom.|date=1997|volume=6|issue=2|pages=247–268}}

{{reflist}}

• A Nowicki, [http://www-users.mat.umk.pl/~anow/ps-dvi/061-fou.pdf the fourteenth problem of hilbert for polynomial derivations]

Category:Differential algebra