A characterization of local nilpotence for dimension two polynomial derivations
aa r X i v : . [ m a t h . A C ] D ec A CHARACTERIZATION OF LOCAL NILPOTENCE FOR DIMENSIONTWO POLYNOMIAL DERIVATIONS
Ivan Pan Abstract.
Let k be an algebraically closed field. We prove that a polynomial k -derivation D in two variables is locally nilpotent if and only if the subgroup of polynomial k -automorphismswhich commute with D admits elements whose degree is arbitrary big. introduction Let k be a field of characteristic 0. A well known result of Rentschler (see [Re1968]) says thata polynomial derivation D : k [ x, y ] → k [ x, y ] (over k ) is locally nilpotent, i.e. for any polynomial f one has D n f = 0 for some n ≥
1, if and only if D is conjugate to a derivation of the form u ( x ) ∂ y , by means of a suitable polynomial automorphism ϕ ∈ Aut k ( k [x , y]). If Aut( D ) denotesthe isotropy subgroup of D with respect to the natural conjugation action of Aut k ( k [x , y]) onthe set Der k ( k [ x, y ]) of k derivations, we deduce that D being locally nilpotent implies Aut( D )is conjugate to a subgroup of the form J u = { ( αx + β, γy + P ( x ); α, γ ∈ k ∗ , β ∈ k , P ∈ k [ x ] , u ( αx + β ) /u ( x ) = γ } , for some u ∈ k [ x ]; here we write ϕ = ( f, g ) ∈ Aut k ( k [x , y]) to mean ϕ ( x ) = f and ϕ ( y ) = g .On the other hand, we consider on Aut k ( k [x , y]) the so-called inductive topology definedby the filtration A ⊂ A ⊂ · · · ⊂ A d ⊂ · · · , where A d = { ( f, g ); deg f, deg g ≤ d } . If k isalgebraically closed, we see that A i is an affine variety for any i and the subgroup above is an infinite dimension algebraic group in the sense of [Sha1981] or [Ka1979]. Following this lastreference, we conclude that Aut( u ( x ) ∂ y ) doesn’t satisfy the property of acting algebraically on k [ x, y ] as an (usual) algebraic group. Indeed, as shown there for a subgroup of Aut k ( k [x , y]) thatproperty is equivalent to being closed (which Aut( D ) does for any derivation D : see [BaPa2019,Cor. 2.2]) and having bounded degree.The aim of the present note is to prove that that holds true exclusively for locally nilpo-tent derivations. More precisely, we have the following characterization of the local nilpotenceproperty whose proof relies strongly on the results of [BaPa2019]. Theorem 1.1.
Let D be a nonzero derivation of k [ x, y ] where k is a field algebraically closed ofcharacteristic 0. Then the following assertions are equivalent: ( a ) D is locally nilpotent; ( b ) for every d there exists ( f, g ) ∈ Aut( D ) such that deg f ≥ d or deg g ≥ d ; ( c ) Aut( D ) is not an algebraic subgroup of Aut( k [ x, y ]) ; ( d ) Aut( D ) is an infinite dimensional algebraic subgroup of Aut( k [ x, y ]) . Research of I. Pan was partially supported by ANII and PEDECIBA, of Uruguay.
Note that whereas ( b ) and ( c ) are equivalent and both are a consequence of ( d ) the converseis not, a priori , necessarily true because Z is not an algebraic subgroup of k . Moreover, sinceconjugating is a homeomorphism which respects the degree’s boundedness, then from Rentchler’sresult we deduce that ( a ) implies ( d ). Thus we only need to prove that ( b ) implies ( a ).Notice also that Theorem 1.1 says that Aut( D ) is an algebraic group when D is not locallynilpotent and then one may ask what kind of such a group corresponds to the conjugation classof a non locally nilpotent derivation.Finally, we observe that Theorem 1.1 doesn’t hold true for dimension 3 or higher (Example2.4) even though assertion ( b ) does for any locally nilpotent derivation D ([BaVe2020, Remark3]). Remark . What we observe here about Aut( D ) recalls what happens with the isotropy ofanother action of Aut k ( k [x , y]). Indeed, it makes one think of the natural action of that groupon the set of reduced principal ideals of height 1 (i.e. algebraic plane curves) where the isotropyof such an ideal ( f ) is not an algebraic group if and only if its generator f ∈ k [ x, y ] may betransformed into an element in k [ x ] by means of an automorphism (see [BlSt2015] and referencestherein). In other words, the ideal generated by such an f would correspond in our research toa locally nilpotent derivation. Moreover, when n > The proof
We denote e (D) the number of D -stable reduced principal ideals of height 1. If ( h ) is sucha principal D -stable ideal, then D ( h ) = λh for some λ ∈ k [ x, y ] and we say h is a eigenvector of D and λ is its (corresponding) eigenvalue . In the case where h is an eigenvector which isreduced, i.e. square-free, we will also say that D stabilizes the curve of equation h = 0. Noticethat e (D) = 0 implies that the kernel ker D of D is equal to k .Two elements f , f ∈ k [ x, y ] are said to be equivalent if there is ϕ ∈ Aut k ( k [x , y]) such that ϕ ( f ) = f . In the case where f is equivalent to x we say it is rectifiable .We keep all notations introduced in the precedent section. If ϕ = ( f, g ) ∈ Aut k ( k [x , y]), wedenote deg ϕ the greatest degree of f and g and call it the degree of ϕ ; notice that we have adegree function deg : Aut k ( k [x , y]) → N which verifies deg ϕψ ≤ deg ϕ deg ψ . Lemma 2.1.
Let D ∈ Der k ( k [ x, y ]) be a nonzero derivation and assume Aut( D ) to be a non-algebraic group. Then one of the following assertions holds ( a ) 0 < e (D) < ∞ , all irreducible eigenvector is rectifiable and D is conjugate to a derivationof the form a∂ x + b∂ y where either ab = 0 and x ∈ k [ x, y ] divides a or a = 0 and b ∈ k [ x ] . ( b ) e (D) = ∞ and D is conjugate to a derivation of the form b ( x ) ∂ y . ( c ) e (D) = ∞ , ker( D ) = k and D stabilizes the members of a pencil of rational curves. ( d ) e (D) = 0 .Proof. We assume e (D) = 0 and prove that one of the assertions ( a ) , ( b ) or ( c ) holds.First suppose 0 < e (D) < ∞ . By [BaPa2019, Thm. A] all irreducible eigenvector is rectifiable.Then there is ϕ ∈ Aut k ( k [x , y]) such that ϕDϕ − = a∂ x + b∂ y admits x ∈ k [ x, y ] as an eigenvector.Hence x divides a . From [BaPa2019, Thm. B] we deduce the assertion ( a ) holds in this case. IMENSION TWO POLYNOMIAL LOCALLY NILPOTENT DERIVATIONS 3
Next suppose e (D) = ∞ . If ker D = k , the references already cited imply we are in thesituation of assertion ( b ). Analogously, if ker D = k , then [BaPa2019, Thms. D] implies we areas in assertion ( c ) which completes the proof. (cid:3) We consider the compactifications X = k ∪ B of k , where B is the union of at most twocurves isomorphic to P : either X = F n , n ≥
1, where F n is the n th Nagata-Hirzebruch surfaceand B is the union of a fiber and the ( − n )-curve in that surface or X = P is the projectiveplane, with B = L ∞ the line at infinity with respect to the affine chart k . We denote byAut( X, B ) the group of automorphisms of X which leave B invariant. Proposition 2.2.
Let D ∈ Der k ( k [ x, y ]) be a nonzero derivation and assume Aut( D ) to be anon-algebraic group. If e (D) = 0 , then D is locally nilpotent.Proof. By Lemma 2.1 we know D satisfies ( a ), ( b ) or ( c ) therein. Moreover, in case (b) theassertion is obvious. Let us consider the case (c), and denote Λ a pencil of rational curves whosemembers are stable under D . From [BaPa2019, Pro. 2.10, Cor. 2.12] we deduce that up toconjugation Aut( D ) may be thought of as either a subgroup of Aut( F n , B ), for some n , or oneof Aut( P , L ∞ ), where in the second case Λ turns out to be a pencil composed by lines passingthrough a point p ∈ L ∞ . Without loss of generality we assume k ⊂ P via the embedding( x, y ) (1 : x : y ), with L ∞ = ( x = 0) and p = (0 : 0 : 1).Since Aut( F n , B ) is an algebraic group then Aut( D ) is necessarily as in the second case, henceAut( D ) is contained in the so-called de Jonqui`eres Group J( k [x , y]) := { ( αx + β, γy + P ( x ); α, β ∈ k ∗ , γ ∈ k , P ∈ k [ x ] } . Moreover, since a general member of Λ corresponds in k to a line of equation x − β = 0, for ageneral β ∈ k , we conclude that if D = a∂ x + b∂ y , then x − β divides D ( x − β ) = a , for any β .Hence a = 0 and D = b ( x, y ) ∂ y . Thus the assertion is consequence of [BaPa2019, Thm. B].Now, assume we are as in the assertion (a) of Lemma 2.1. Up to conjugation we may assume D = x ℓ a ( x, y ) ∂ x + b ( x, y ) ∂ y , where we may suppose a = 0 and x does not divide a because a = 0and b ∈ k [ x ] leads to the required conclusion.We have a homomorphism Aut( D ) → Per(E), where E is the set of prime principal ideals of k [ x, y ] which are D -stable and Per(E) denotes the finite group of permutations of E . Hence theprincipal ideal x k [ x, y ] belongs to E . Since the degree function deg : Aut k ( k [x , y]) → N is notbounded on Aut( D ) we deduce it is not bounded on the kernel K of that homomorphism, so K is not an algebraic group. Note that an element ϕ = ( f, g ) ∈ K verifies ϕ ( x ) /x ∈ k ∗ , hence f = αx for some α ∈ k ∗ . Since the jacobian of ϕ is constant we deduce g = γy + P ( x ) for some γ ∈ k ∗ and P ∈ k [ x ]. In other words K is contained in the subgroup of J( k [x , y]) whose elementsfix the ideal generated by x . More explicitly, if ϕ = ( αx, γy + P ( x )) ∈ K , then (cid:26) α ℓ − a ( αx, γy + P ( x )) = a ( x, y ) b ( αx, γy + P ( x )) = γb ( x, y ) + x ℓ a ( x, y ) P ′ ( x ) . (1)On the other hand, since K is not an algebraic group we deduce it contains a sequence ( ϕ n ) n ≥ of elements such that the corresponding sequence of degrees (deg ϕ n ) n ≥ is increasing. We willshow in several steps this implies a = 0 which yields a contradiction and terminates the proof.Write ϕ n = ( α n x, γ n y + P n ( x )), n ≥ DIMENSION TWO POLYNOMIAL LOCALLY NILPOTENT DERIVATIONS
First we observe that a does not depend on y . Indeed, write a = P di =0 a i ( x ) y i , d ≥ P n increases with n , if d > P n ( x ) d isbounded, hence d = 0.Second, by an analogous reasoning the bottom equality in (1) gives b = b ( x ) + b ( x ) y , where b = 0, and then that equality is equivalent to the following two ones b ( αx ) + b ( αx ) P ( x ) = γb ( x ) + x ℓ a ( x ) P ′ ( x ) , (2) b ( αx ) = b ( x ) . (3)Now write a = r X i =0 A i x i , b = s X i =0 B i x i , where A r B s = 0. If P = P n = P mi =0 p i x i for some n ≫
0, we deduce m = m n = α sn B s A − r . Hence we may suppose α sn = 1 because m n increases with n , and so s = deg b >
0. From (3)we deduce α = 1, a contradiction which finishes the proof. (cid:3) Now we treat the case ( d ) of Lemma 2.1. We have the following result valid over an arbitraryfield of characteristic zero which together with Proposition 2.2 readily leads Theorem 1.1. Proposition 2.3.
Let D be a derivation such that e (D) = 0 . Then Aut( D ) is finite.Proof. If k = C is the field of complex numbers, the result is a straightforward consequence of[CMP2019, Thm. A]. Denote by ℓ the order of Aut( D ) in that case. It suffices to prove that if F = { ϕ , . . . , ϕ n } is a subset of Aut( D ) in the general case, then n ≤ ℓ .In fact, inspired by the proof of [DeKa2009, Prop. 1.4] we consider the extension k of Q obtained by adjoining the coefficients of D ( x ) , D ( y ), ϕ i ( x ), ϕ i ( y ), i = . . . , n . Then D and all ofthe ϕ ′ i s restraint to give a derivation and suitable automorphisms D , ϕ i : k [ x, y ] → k [ x, y ], i = 1 . . . , n , such that ϕ i D = Dϕ i for all i . Since k is isomorphic to a subfield of C we maysuppose k ⊂ C , and then all these maps extend to C [ x, y ] from which the assertion follows. (cid:3) Example . Theorem 1.1 doesn’t hold true for n >
2. Indeed, let D = P ni =1 a i ∂ x i be aderivation of B = k [ x , . . . , x n ], where a , . . . , a n − don’t depend on x , x n and a n = 0. Then D induces a derivation in A = k [ x , . . . , x n − ]. If a , . . . , a n − are general enough to ensure D isnot locally nilpotent as derivation in A , then it is so also as derivation in B . However, Aut( D )contains the automorphisms of the form( x + p ( x n ) , . . . , x n − , x n ) , p ∈ k [ x n ] , hence it contains elements defined by polynomials of arbitrary degree. References [BaPa2019] R. Baltazar and I. Pan,
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