ON THE RING OF CONSTANTS FOR DERIVATIONS OF POWER SERIES RINGS IN TWO VARIABLES

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1 ON THE RING OF CONSTANTS FOR DERIVATIONS OF POWER SERIES RINGS IN TWO VARIABLES BY LEONID MAKAR - LIMANOV DETROIT) AND ANDRZEJ NOWICKI TORUŃ) Abstract. Let k[[x, y]] be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k[[x, y]]. We prove that if Kerd) k then Kerd) =Kerδ), where δ is a jacobian derivation of k[[x, y]]. Moreover, we prove that Kerd) is of the form k[[h]], for some h k[[x, y]]. 1. Introduction. Let k[x, y] be the ring of polynomials in two variables over a field k of characteristic zero. Let d be a nonzero derivation of k[x, y] and let A be its ring of constants. It is well known see for example [4]) that if A k then A = Kerδ), where δ is a jacobian derivation of k[x, y]. It is also well known [5] or [4]) that A is of the form k[h] for some h k[x, y]. In 1975, A. P loski [6]) proved similar facts for derivations in convergent power series ring in two variables. In this paper we show that the above facts are also true for derivations in formal power series ring in two variables. 2. Preliminaries. If F is a nonzero power series from k[[x, y]] then we denote by ωf ) the lowest homogeneous form of F, and we denote by of ) the order of F, that is, of ) = deg ωf ). Moreover, we assume that ω0) = 0 and o0) =. It is clear that of G) = of ) + og) for all F, G k[[x, y]]. By a derivation of k[[x, y]] we mean every k-linear mapping d : k[[x, y] k[[x, y]] such that df G) = F dg)+gdf ) for F, G k[[x, y]]. Let us recall see for example [2], [4]) that each derivation d of k[[x, y]] has a unique presentation of the form d = F + G, where F, G k[[x, y]]. Recall also that if d is a derivation of k[[x, y]] then its kernel k[[x, y]] d = {F k[[x, y]]; df ) = 0} is a subring of k[[x, y]] containing k. If F, G are two power series from k[[x, y]] then we denote by JF, G) the jacobian of F, G, that is, JF, G) = F G F G. If F k[[x, y]] is fixed then a mapping δ : k[[x, y]] k[[x, y] defined as δg) = JF, G), for G k[[x, y]], is a derivation of k[[x, y]]; we call it a jacobian derivation Mathematics Subject Classification: Primary 12H05, Secondary 13F25 Supported by NSF Grant DMS and KBN Grant 2 PO3A

2 Note the following useful lemma. Lemma 2.1. Let f, g be nonzero homogeneous polynomials from k[x, y]. Assume that degf) = sn, degg) = sm, where gcdn, m) = 1. If Jf, g) = 0 then there exist a homogeneous polynomial h k[x, y] of degree s and nonzero elements a, b k such that f = ah n and g = bh m. Proof. See, for example, [1]. 3. Jacobian derivations In this section we prove the following Theorem 3.2. Let d be a nonzero derivation of k[[x, y]]. If F k[[x, y]] d k, then k[[x, y]] d = k[[x, y]] δ, where δ = JF, ). For the proof of this theorem we need two lemmas. Lemma 3.3. Let d be a nonzero derivation of k[[x, y]]. If F, G k[[x, y]] d then the polynomials ωf ), ωg) are algebraically dependent over k. Proof. It is obvious when dx) = 0 or dy) = 0 or ωf ) k or ωg) k. So we may assume that dx) 0, dy) 0, deg ωf ) 1 and deg ωg) 1. Put dx) = U, dy) = V, and let F = F ωf ), G = G ωg), U = U ωu), V = V ωv ). Then: 1) 0 = df ) = + 0 = dg) = + ωg) ωg) ) ωu) ) + F + U + F ) ωv ) + V ), ) ωu) ) + G + U + G ) ωv ) + V ). Assume that deg ωu) < deg ωv ). Then comparing the lowest forms in 1) we have: ωu) = 0 and ωg) ωu) = 0. = 0 and the polynomials ωf ) and ωg) are algebraically dependent because they belong to k[y]. If deg ωu) > deg ωv ) then analogously ωf ) and ωg) are in k[x] and are also dependent. Assume now that deg ωu) = deg ωv ). Then comparing the lowest forms in 1) we get the following system of equations: Hence = ωg) ωu) + ωv ) = 0, ωg) ωg) ωu) + ωv ) = 0. Since ωu), ωv )) 0, 0), this system has a nonzero solution. This means that the jacobian JωF ), ωg)) is equal to zero and so, the polynomials ωf ) and ωg) are algebraically dependent. 2

3 Lemma 3.4. Let d 1 and d 2 be nonzero derivations of k[[x, y]]. Assume that the rings k[[x, y]] d1 and k[[x, y]] d2 both contain a series F k[[x, y]] k. Then k[[x, y]] d1 = k[[x, y]] d2. Proof. We can assume that ωf ) k. By Lemmas 3.3 and 2.1 we know that ωf ) is a polynomial of h 1 and a polynomial of h 2 for some h 1, h 2 k[x, y]). So we may assume that h 1 = h 2 = h. Let us take any G k[[x, y]] for which ωg) is not a polynomial of h. Then d 1 G) 0 and d 2 G) 0. Let us consider the derivation d 3 = d 2 G)d 1 d 1 G)d 2. It is clear that d 3 F ) = d 3 G) = 0. But ωf ) and ωg) are algebraically independent. By Lemma 3.3 it is possible only if d 3 = 0. So d 2 G)d 1 = d 1 G)d 2 and the kernels are the same. Proof of Theorem 3.2. It is a simple consequence of Lemma 3.4 because d 0, δ 0 and the rings k[[x, y]] d and k[[x, y]] δ contain F k[[x, y]] k. 4. A generator of the ring of constants Let d : k[[x, y]] k[[x, y]] be a nonzero derivation and let A = k[[x, y]] d. We want to show that the ring A is of the form k[[f ]] for some series F k[[x, y]]. If A = k, then A = k[[f ]] for F = 0. Let us assume that A k. We already know by Lemmas 3.3 and 2.1) that all lowest homogeneous forms of nonzero elements in A are scalar multiples of powers of a homogeneous form ϕ. For each F A {0} let us denote by γf ) the degree of ϕ in ωf ), that is, if ωf ) = aϕ n where 0 a k, then γf ) = n. Assume moreover that γ0) =. Let us consider the semi-group π = {γf ); 0 F A}. Since A k this semi-group contains positive numbers. Let γ be the greatest common divisor of the elements of π. Lemma 4.5. There exist F, G A {0} such that γ = γf ) γg). Proof. Since γ is the greatest common divisor, there exist nonnegative integers i 1,..., i n, j 1,..., j m and nonzero series F 1,..., F n, G 1,..., G m from A such that Put F = F i1 1 F n in γf ) γg). γ = i 1 γf 1 ) + + i n γf n ) j 1 γg 1 ) j m γg m ). and G = G j1 1 Gjm m. Then F, G A {0} and γ = Lemma 4.6. Let F, G A {0} be as in Lemma 4.5. Let γg) = sγ. Then nγ π, for any n > s 2 s 1. 3

4 Proof. Since γ = γf ) γg), we have γf ) = s + 1)γ. Assume that n > s 2 s 1. Let n = us + r, where u, r are integers such that 0 r < s. Put i = r, j = u r. Then i 0 and j 0 since u s 1 r), and moreover is + 1) + js = rs + 1) + u r)s = us + r = n. Let H = F i G j. Then 0 H A and that is, nγ π. γh) = iγf ) + jγg) = is + 1)γ + jsγ = nγ, We may extend the mapping γ ) to A 0 {0} where A 0 is the field of fractions of A) by defining for all nonzero A, B A. γa/b) = γa) γb) Lemma 4.7. Let f, g A 0 {0}. If γf) = γg), then there exists c k {0} such that γf cg) > γf). Proof. It is clear if f, g A. Let f = A/B, g = C/D, where A, B, C, D A {0}. Since γf) = γg), we have γad) = γa) + γd) = γc) + γb) = γcb), and so, there exists nonzero c k such that γad ccb) > γad). Then we have γf cg) = γad ccb)/bd) = γad ccb) γbd) > γad) γbd) = γa/b) = γf), that is γf cg) > γf). Consider now the fraction h = F/G, where F and G are such nonzero series from A as in Lemma 4.5. We know that γh) = γ. We want to show that h A. Lemma 4.8. There exists a natural number n such that h n A. Proof. Let γg) = sγ, γf ) = s + 1)γ and let n be a natural number such that n > s 2 s. We shall show that h n k[[x, y]]. We know, by Lemma 4.6 and its proof, that there exist integers i 1 0, j 1 0 such that γh 1 ) = nγ, where H 1 = F i1 G j1. Then γh n ) = nγ = γh 1 ), so by Lemma 4.7) γh n c 1 H 1 ) > γh n ) for some c 1 k {0}. 4

5 Put h 1 = h n c 1 H 1. If h 1 = 0 then h n = c 1 H 1 A and we are done. Assume that h 1 0. Then γh 1 ) = n 1 γ, where n 1 > n n 0. Using again Lemmas 4.6 and 4.7 we see that γh 1 c 2 H 2 ) > γh 1 ), for some c 2 k {0}, H 2 = F i2 G j2, i 2 0, j 2 0. Put h 2 = h 1 c 2 H 2 = h n c 1 H 1 c 2 H 2. If h 2 = 0, then h n A and we are done, and so on. If the above procedure has no end, then we obtain an infinite sequence c m H m ) of nonzero elements from A such that oh m ) < oh m+1 ) for any natural m and oh n U m+1 ) > oh n U m ), where U m = c 1 H 1 + +c m H m. This means that h n is the limit of the convergent sequence U m ). Since each U m belongs to the ring k[[x, y]] which is complete, the limit h n also belongs to k[[x, y]]. Therefore h n k[[x, y]] A 0 = A. Lemma 4.9. h k[[x, y]]. Proof. The ring k[[x, y]] is a unique factorization domain see, for example, [3] p.163), hence it is integrally closed. Lemma 4.8 implies that h integral over k[[x, y]], so h k[[x, y]]. Lemma A = k[[h]]. Proof. We already know by the previous lemma) that h k[[x, y]] A 0 = A. We know also that γh) = γ 1, so oh) 1 that is h has no constant term). It is clear that k[[h]] A. Let U A k. We shall show that U k[[h]]. Since γu) = n 1 γ for some natural n 1, there exists a nonzero c 1 k such that γu 1 ) > γu) for U 1 = U c 1 h n1. If U 1 = 0 then U k[[h]] and we are done. Assume that U 1 0. Since U 1 A there exist n 2 > n 1 and 0 c 2 k such that γu 2 ) > γu 1 ) for U 2 = U 2 c 2 h n2 = U c 1 h n1 c 2 h n2. If U 2 = 0 then U k[[h]] and we are done, and so on. If the above procedure has an end we see that U k[[h]]. In the opposite case U is the limit of the infinite convergent sequence h m ), where each h m is of the form h m = c 1 h n1 + + c m h nm, where c 1,..., c m are nonzero elements of k and n 1 < < n m. Therefore U k[[h]]. From the above lemmas we get the following main result of our paper. Theorem If d is a nonzero derivation of k[[x, y]], then k[[x, y]] d = k[[h]] for some h k[[x, y]]. References [1] H. Appelgate, H. Onishi, The jacobian conjecture in two variables, J. Pure Applied Algebra, ),

6 [2] N. Bourbaki, Éléments de Mathématique, Algébre, Livre II, Hermann, Paris, [3] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, [4] A. Nowicki, Polynomial derivations and their rings of constants, N. Copernicus University Press, Toruń, [5] A. Nowicki, M. Nagata, Rings of constants for k-derivations in k[x 1,..., x n ], J. Math. Kyoto Univ., ), [6] A. P loski, On the jacobian dependence of power series, Bull. Acad. Pol. Sci., ), Department of Mathematics and Computer Science Bar-Ilan University Ramat-Gan, Israel, lml@macs.biu.ac.il); Department of Mathematics, Wayne State University Detroit, Michigan 48202, USA, lml@bmath.wayne.edu) N. Copernicus University, Faculty of Mathematics and Informatics, Toruń, POLAND, anow@mat.uni.torun.pl). 6

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