Abstract. In this short note we propose a new very easy and elementary proof of the known fact that every triangular automorphism of kn is the ex-ponent of a suitably chosen locally nilpotent k-derivation on k[x1,..., xn]. Two other, different proofs of this fact can be found in [2] and [3]. 1. Introduction. Le

Karaś, Marek

Acta Mathematica

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Jagiellonian Univeristy Repository

UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLVI2008A NOTE ON TRIANGULAR AUTOMORPHISMSby Marek KaraśAbstract. In this short note we propose a new very easy and elementaryproof of the known fact that every triangular automorphism of kn is the ex-ponent of a suitably chosen locally nilpotent k-derivation on k[x1, . . . , xn].Two other, different proofs of this fact can be found in [2] and [3].1. Introduction. Let k be a field of characteristic zero and R a k-algebra.Recall that a k-derivation on R is a k-linear map D : R → R satisfying theLeibniz rule: D(ab) = aD(b) + bD(a) for all a, b ∈ R. A derivation D on aring R is called locally nilpotent if for every a ∈ R there is an n ∈ N suchthat Dn(a) = 0. If D : R → R is a locally nilpotent k-derivation, then themapping exp D : R → R given by the formula exp D(a) =∑∞i=01i!Di(a) is ak-automorphism of R (see e.g. [2] or [4]).Recall also that a k-automorphism F : k[X1, . . . , Xn] → k[X1, . . . , Xn] ofthe polynomial ring in n variables X1, . . . , Xn over a field k is called triangularif F (Xi) = Xi + fi(X1, . . . , Xi−1), for i = 1, . . . , n. Since k is an infinite field,there is an isomorphism between the group of the ring k[X1, . . . , Xn] and thering of polynomial automorphism of kn, given by the formula G 7→ G∗ =(G(X1), . . . , G(Xn)).In this short note we give an easy proof of the following theorem, whichhas already been proved (see [1] and [3]).2000 Mathematics Subject Classification. 14Rxx, 14R10.Key words and phrases. Triangular automorphism, locally nilpotent derivation, exponentautomorphism.70Theorem 1.1. For all n > 1 and for all polynomials f1 ∈ k, f2 ∈ k[X1],f3 ∈ k[X1, X2], . . . , fn ∈ k[X1, . . . , Xn−1] there exists a locally nilpotent k-derivation D : k[X1, . . . , Xn]→ k[X1, . . . , Xn] such that(exp D)∗ :x1x2...xn 7→x1 + f1x2 + f2(x1)...xn + fn(x1, . . . , xn−1) .The proof of the above theorem can also be found in [1] and [3]. Theproof given in [1] uses the Campbell–Hausdorff formula for exp D1 ◦ exp D2,and the one given in [3] uses the notion of the logarithm of locally nilpotentmap E : kn → kn (more precisely, the logarithm of idkn + E). Our proof iscompletely different and perhaps easier.An easy consequence of Theorem 1.1, also already known, is the followingCorollary 1.2. If F : kn → kn is a polynomial automorphism of theformF :x1x2...xn 7→a1x1 + f1a2x2 + f2(x1)...anxn + fn(x1, . . . , xn−1) ,where a1, . . . , an ∈ k\{0}, then there exists a locally nilpotent derivation D :k[x1, . . . , xn] → k[x1, . . . , xn] such that F = (exp D)∗ ◦ L, where L : kn → knis linear with the diagonal matrix determined by a1, . . . , an.Proof. F ◦ L−1 is of the triangular form. Following Theorem 1.1 thereexists a locally nilpotent k-derivation D : k[X1, . . . , Xn]→ k[X1, . . . , Xn] suchthat F ◦ L−1 = (exp D)∗ .2. Proof. We start with the following lemmaLemma 2.1. Let R be a k-algebra and D : R → R be a locally nilpo-tent k-derivation such that for every g ∈ R there is g̃ ∈ R such that g =∑∞i=11i!Di−1(g̃). For an f ∈ R define the k-derivation D̃ : R[t] → R[t] on thepolynomial ring in one variable t over R such that D̃|R = D and D̃(t) = f̃ ,where f̃ ∈ R is such that f =∑∞i=11i!Di−1(f̃). Then(1) D̃ is locally nilpotent,(2) exp D̃|R = exp D and(exp D̃)(t) = t + f,(3) for every h ∈ R[t] there is h̃ ∈ R[t] such that h =∑∞i=11i!D̃i−1(h̃).71Proof. Assertions that D̃ is locally nilpotent and exp D̃|R = exp D areobvious. Moreover,(exp D̃)(t) =∞∑i=01i!D̃i(t) = D̃0(t) +∞∑i=11i!Di−1(D̃(t))= t +∞∑i=11i!Di−1(f̃) = t + f.Statement (3) will be proved by induction with respect to k = degt g. Ifk = 0, i.e., g ∈ R, then, by the assumptions, there exists an element g̃ ∈ R ⊂R[t] such that g =∑∞i=11i!Di−1(g̃) =∑∞i=11i!D̃i−1(g̃). Now assume that (3) istrue for k ≥ 0 and consider a polynomial:g = ak+1tk+1 + aktk + . . . + a0,where ak+1, ak, . . . , a0 ∈ R.By the assumptions, there is an element bk+1 ∈ R such that ak+1 =∑∞i=11i!Di−1(bk+1). Denote g̃1 = bk+1tk+1, g1 =∑∞i=11i!D̃i−1(g̃1) and observethatg1 =[ ∞∑i=11i!Di−1(bk+1)]tk+1 + . . . = ak+1tk+1 + . . .Indeed, for all l ≥ 0 there is:D̃l(bk+1tk+1) =l∑i=0(li)D̃i(bk+1)D̃l−i(tk+1) =l∑i=0(li)Di(bk+1)D̃l−i(tk+1)= Dl(bk+1)tk+1 +l−1∑i=0(li)Di(bk+1)D̃l−i(tk+1).Since degt D̃(h) < degt h for each h ∈ R[t]\R, we see thatdegt D̃j(tk+1) < k + 1for j > 0.Thus degt(g− g1) < degt g, and, by the induction assumption, there existsg̃2 ∈ R[t] such that:g − g1 =∞∑i=11i!D̃i−1(g̃2).Putting g̃ = g̃1 + g̃2 we obtain∞∑i=11i!D̃i−12 (g̃) =∞∑i=11i!D̃i−12 (g̃1) +∞∑i=11i!D̃i−12 (g̃2) = g1 + (g − g1) = g.72Proof of Theorem 1.1. Consider the k-derivation D0 = 0 on k. SinceDj0(h) = 0 for all h ∈ k and j > 0, then exp D0 = idk and:∞∑i=11i!Di−11 (h) = hfor all h ∈ k.Applying Lemma 2.1 for R = k, D = D0 and f = f1, we obtain the locallynilpotent k-derivation D1 : k[X1]→ k[X1] such that(exp D1)∗ :{x1}7→{x1 + f1}and that for every h ∈ k[X1] there is h̃ ∈ k[X1] such that h =∑∞i=11i!Di−11 (h̃).Thus we can apply Lemma 2.1 for R = k[X1], D = D1 and f = f2. In this waywe obtain the locally nilpotent k-derivation D2 : k[X1, X2] → k[X1, X2] suchthat(exp D2)∗ :{x1x2}7→{x1 + f1x2 + f2(x1)}and that for every h ∈k[X1, X2] there is h̃ ∈k[X1, X2] with h =∑∞i=11i!Di−12 (h̃).Now it is easy to see that applying Lemma 2.1 n times, we complete theproof of Theorem 1.1References1. Drensky V., Yu J.-T., Exponential automorphism of polynomial algebras, Comm. Algebra,26 (1998), 2977–2985.2. van den Essen A., Polynomial automorphism and the Jacobian Conjecture, BirkhäuserVerlag, Basel–Boston–Berlin, 2000.3. Freudenburg G., Algebraic theory of locally nilpotent derivation, Springer-Verlag, Berlin–Heidelberg, 2006.4. Nowicki A., Polynomial derivations and their rings of constants, Univ. of Toruń, Toruń,1994.Received July 15, 2008Institute of MathematicsJagiellonian Universityul.  Lojasiewicza 630-348 Kraków, Polande-mail : Marek.Karas@im.uj.edu.pl

A note on triangular automorphisms

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