Rings of constants of linear derivations on Fermat rings

arXiv:1802.10578v1 [math.AC] 28 Feb 2018 Rings of constants of linear derivations on Fermat rings Marcelo Veloso e-mail: veloso@ufsj.edu.br Ivan Shestakov e-mail: shestak@ime.usp.br Abstract In this paper we characterize all the linear C-derivations of the Fermat ring. We show that the Fermat ring has linear C-derivations with trivial ring of constants and construct some examples. Keywords: Derivations, Fermat ring, ring of constants. 2010 AMS MSC: 13N15, 13A50, 16W25. Introduction The present paper deals with C-derivations of the Fermat ring Bnm = C[X1 , . . . , Xn ] , (X1m1 + · · · + Xnmn ) where C[X1 , . . . , Xn ] is the polynomial ring in n variables over the complex numbers C, n ≥ 3, m = (m1 , . . . , mn ), mi ∈ Z and mi ≥ 2 for i = 1, . . . , n. It is well known the difficulty to describe the ring of constants of an arbitrary derivation (see [1, 3, 4, 5]). It is also difficult to decide if the ring of constants of a derivation is trivial (see [5, 6, 7]). In this work we study the ring of constants of linear derivations of Fermat rings and its locally nilpotent derivations. In [5], Andrzej Nowicki presents a description of all linear C-derivations of the polynomial ring C[X1 , . . . , Xn ] which do not admit any nontrivial rational constant. In a recent paper [1], P. Brumatti and M. Veloso show that for m = (2, . . . , 2) the ring Bnm has nonzero irreducible locally nilpotent derivations. Furthermore, whenever m1 = · · · = mn , They show that certain classes of derivations of C[X1 , . . . , Xn ] do not induce derivations of Bnm or are not locally nilpotent if they do. In this work we obtain some similar results to [1], considering more general Fermat rings. We present a description of all the linear C-derivations of Bnm 1 when m = (m1 , . . . , mn ) and mi ≥ 3 (Theorem 5) and m = (2, . . . , 2) (Theorem 6). We also provide examples of linear derivations of Bnm with trival ring constants. The text is organized as follows: Section 1 gathers the basic definitions and notations. Further, we discuss several properties of the ring Bnm are discussed and a set of generators for Der(Bnm ) is presented. Section 2 is dedicated to the study of the linear derivations of the Fermat ring. The set of all locally nilpotent C-derivations of Bnm is studied in Section 3. Finally, Sections 4 and 5 are devoted to the study of the rings of constants of linear derivations of Fermat rings. 1 Preliminaries and Some Results In this paper the word “ring" means a commutative ring with unit and characteristic zero. Furthermore, we denote the group of units of a ring R by R∗ and the polynomial ring in n variables over R by R[X1 , . . . , Xn ]. A “domain” is an integral domain. An additive mapping D : R → R is said to be a derivation of R if it satisfies the Leibniz rule: D(ab) = aD(b) + D(a)b, for all a, b ∈ R. If A is a subring of R and D is a derivation of R satisfying D(A) = 0 is called D an A-derivation. The set of all derivations of R is denoted by Der(R), the set of all A-derivations of R by DerA (R) and by ker(D), the ring of constants of D, that is ker(D) = {a ∈ R | D(r) = 0}. In this paper, the word "derivation" implicitly means a derivation which is C-derivation and therefore we will use the notation Der(Bnm ) to denote DerC (Bnm ). The residue classes of variables X, Y , Z, ... module an ideal are represented by x, y, z, respectively. The symbol C is reserved to indicate the field of complex numbers. A derivation D is locally nilpotent if for each r ∈ R there is an integer n ≥ 0 such that Dn (r) = 0. We denote by LN D(R) the set of all locally nilpotent derivations of R. We say that a element b ∈ R is a Darboux element of D ∈ Der(R) if b 6= 0, b is not invertible in R and D(b) = λb for some λ ∈ R. In other words, a nonzero element b of R is a Darboux element of D if, and only if, the principal ideal (b) = {rb | r ∈ R} is different from R and it is invariant with respect to D, that is D((b)) ⊂ (b). If b is a Darboux element of D, then every λ ∈ R, such that D(b) = λb, is said to be an eigenvalue of b. In particular, every element nonzero and noninvertible element belonging to the ring of constants, ker D, is a Darboux element of D. If R is a domain and D(b) = λb, then it is easy to see such the eigenvalue λ is unique. Lemma 1 Let Bnm where m = (m1 , . . . , mn ). Then, for each f ∈ Bnm , there is a unique F ∈ C[X1 , . . . , Xn ] such that degXn < mn and f = F (x1 , . . . , xn ). Proof. It follows directly from the Euclidean division algorithm by considering the polynomial X1m1 + · · · + Xnmn as a monic polynomial in Xn with coefficients 2 in C[X1 , . . . , Xn−1 ]. ♦ Theorem 2 ([3, Theorem 4]) If n ≥ 5 and mi ≥ 2 for all 1 ≤ i ≤ n, then Bnm is a unique factorization domain. ♦ mn 1 We also can write Bnm = C[x1 , . . . , xn ], where xm = 0. Here 1 + · · · + xn x1 , x2 , . . . , xn are the images of X1 , X2 , . . . , Xn under the canonical epimormn−1 1 phism C[X1 , . . . , Xn ] → Bnm . An element of form axm or 1 · · · xn−1 mn−1 j m1 bx1 · · · xn1 xn , for 1 ≤ j ≤ mn − 1, is called monomial . A nonzero element f ∈ C[x1 , . . . , xn ] is said to be homogeneous element of degree k if f is of the form X f= a(i1 ···in ) xi11 · · · xinn i1 +···+in =k where 1 ≤ in ≤ mn − 1 and a(i1 ···in ) ∈ C for all (i1 · · · in ). We assume that the zero element is a homogeneous element of any degree. Furthermore, we denote by Vk the set of all homogeneous elements of degree k. Clearly Vk is a subspace of Bnm . 1.1 A set of generators for Der(Bnm ) Now we will present a set of generators for the Bnm -module Der(Bnm ). First some notation will be established. Given H ∈ S = C[X1 , . . . , Xn ] and 1 ≤ i ≤ n, the partial derivative ∂(H) ∂Xi is denoted by HXi . For all pairs ∂ ∂ −HXj i, j ∈ {1, . . . , n} with i 6= j, we define the derivation DHij = HXi ∂Xj ∂Xi on S. Observe that DHij (H) = 0. [n] Let A = CI be a finitely generated C-algebra. Consider the C[n] -submodule DI = {D ∈ DerC (C[n] ) | D(I) ⊆ I} of the module DerC (C[n] ). It is well known that the C[n] -homomorfism ϕ : DI → DerC (A) given by ϕ(D)(g + I) = D(g) + I induces a C[n] -isomorfism of IDerDC I(C[n] ) in DerC (A). The Theorem 3 will be needed, its proof can be found in [2, P roposition 1]. Theorem 3 Let F ∈ C[n] = C[X1 , . . . , Xn ] (n ≥ 2) be such that {FX1 , ..., FXn } is a regular sequence in S. If there exists a derivation ∂ on S such that ∂(F ) = αF for some α ∈ C, then the C[n] -module DF := {D ∈ Der(S) | D(F ) ∈ F · C[n] } F is generated by the derivation ∂ and the derivations Dij = Dij for i < j. From now on, the derivations DFij , where F = X1m1 + · · · + Xnmn , will be denoted by Dij. Since  m −1 if k=i  −mj Xj j Dij (Xk ) = mi Ximi −1 if k=j  0 if k ∈ / {i, j} 3 so Dij (F ) = 0. Then Dij ∈ Der(S) induces dij = mi ximi −1 in Der(Bnm ). Consider the derivation E= ∂ m −1 ∂ − mj xj j ∂xj ∂xi ∂ ∂ 1 1 X1 + ··· + Xn . m1 ∂X1 mn ∂Xn Note that E satisfies E(F ) = F . Hence, E ∈ Der(S) induces ε= 1 ∂ ∂ 1 x1 + ··· + xn ∈ Der(Bnm ) m1 ∂x1 mn ∂xn As a consequence of Theorem 3 the following result is obtained: Proposition 4 If F = X1m1 + · · · + Xnmn then DF := {D ∈ Der(S) | D(F ) ∈ F · S} is generated by the derivation E and the derivations Dij , i < j. In particular the Bnm -module Der(Bnm ) is generated by the derivation ε and by the derivations dij , for i < j. Proof. Since {m1 X m1 −1 , . . . , mn X mn −1 } is a regular sequence and E(F ) = F the result following by Theorem 3. ♦ 2 Linear derivations This section is dedicated to the study of the linear derivations of the Fermat ring Bnm = C[x1 , . . . , xn ], mn 1 where xm 1 + · · · + xn = 0. A derivation d of the ring Bnm is called linear if d(xi ) = n X aij xj for i = 1, . . . , n, where aij ∈ C. j=1 The matrix [d] = [aij ] is called the associated matrix of the derivation d. Theorem 5 Let d ∈ Der(Bnm ) be linear. If m = (m1 , . . . , mn ) with mi ≥ 3 for all i = 1, . . . , n, then its associated matrix [d] is a diagonal matrix and has the following form   α m1 for some α ∈ C.     α m2 .. . α mn 4   .  Proof. Let [d] = [aij ] be the associated matrix of d. Then d(xi ) = n X aij xj , for j=1 mn 1 all i. Since xm 1 + · · · + xn = 0, m1 x1m1 −1 d(x1 ) + · · · + mn xnmi −1 d(xn ) = 0. Then, n n n X X X n −1 anj xj ) ( a2j xj ) + · · · + mn xm a1j xj ) + m2 x2m2 −1 ( 0 = m1 x1m1 −1 ( n j=1 j=1 j=1 (2.1) Now note that n n X X 1 a1j xj ) =m1 a11 (xm ) + m a1j xj x1m1 −1 m1 x1m1 −1 ( 1 1 j6=1 j=1 mn 2 =m1 a11 (−xm 2 − · · · − xn ) + m1 n X a1j xj x1m1 −1 j6=1 and n n X X 2 a2j xj ) = m2 a22 xm + m a2j xj x2m2 −1 m2 x2m2 −1 ( 2 2 j6=2 j=1 .. . n n X X mn n −1 n −1 + m a x ) = m a x anj xj xm mn xm ( nj j n nn n n n n j6=n j=1 replacing in the Equation (2.1) we obtain mn 2 0 = (m2 a22 − m1 a11 )xm 2 + · · · + (mn ann − m1 a11 )xn + m1 n X a1j xj x1m1 −1 + j6=1 m2 n X a2j xj x2m2 −1 + · · · + mn j6=2 n X n −1 anj xj xm . n j6=n (2.2) Observe that if mi ≥ 3, then mi −1 mn 1 {xm | 1 ≤ i < j ≤ n, } ∪ {xj ximi −1 | 1 ≤ j < i ≤ n} 2 , . . . , xn } ∪ {xj xi is a linearly independent set over C. Thus, we conclude that mn ann = · · · = m2 a22 = m1 a11 = α and aij = 0 if i 6= j, 5 i.e. aij =  if if 0 α mi i 6= j i=j ♦ This theorem shows that for m = (m1 , . . . , mn ) and mi ≥ 3 linear derivations of B2m are what is called diagonal derivations. The next result characterizes linear derivations of Bnm whenever m = (2, . . . , 2). Previously, remember that a square matrix with complex elements A is said to be skew-symmetric matrix if AT = −A (here AT stands, of course, for the transpose of the matrix A). Theorem 6 Let d ∈ Der(Bnm ) be linear. If m = (2, . . . , 2), then there exist a scalar derivation dα ([dα ] is a scalar matrix) and a skew-symmetric derivation ds ([ds ] a skew-symmetric matrix) such that d = dα + ds . This decomposition is unique. Proof. Let d ∈ Der(Bnm ) be a linear derivation and A = [aij ] its associated matrix. Using the same arguments used in Theorem 5 we obtain X (aij + aji )xi xj 0 = (a22 − a11 )x22 + · · · + (ann − a11 )x2n + i<j {x22 , . . . , x2n } Since the set C, it follows that ∪ {xi xj ; 1 ≤ i < j ≤ n} is linearly independent over a11 = a22 = · · · = ann = α and aij = −aji if i < j, then its associated matrix [d] has the following form   α a12 . . . a1n  −a12 α a2n     .. .. ..  . . .  . . . .  −a1n −a2n . . . α where α, aij ∈ C. Now define dα by dα (xi ) = αxi , i = 1, . . . , n and ds = d − dα . ♦ 3 Locally Nilpotent Derivations In this section we proof that the unique locally nilpotent derivation linear of Bnm for m = (m1 , . . . , mn ) and mi ≥ 3 is the zero derivation. Further, we show that a certain class of derivations of C[X1 , . . . , Xn ] do not induce nonzero locally nilpotent derivation of Bnm . [n] Let S = CI be a finitely generated C-algebra. Consider the C[n] -submodule DI = {D ∈ DerC (C[n] ) | D(I) ⊆ I} of the module DerC (C[n] ). It is well known 6 that the C[n] -homomorfism ϕ : DI → DerC (S) given by ϕ(D)(g + I) = D(g) + I induces a I C[n] -isomorfism of IDerDC (C in Der (S). From this fact the following result is C [n] ) obtained. Proposition 7 Let d be a derivation of the Bnm . If d(x1 ) = a ∈ C and for each i, 1 < i ≤ n, d(xi ) ∈ C[x1 , . . . , xi−1 ], then d is the zero derivation. Proof. Let F be the polynomial X1m1 + · · · + Xnmn . We know that exists D ∈ Der(C[n] ) such that D(F ) ∈ F C[n] and that d(xi ) = D(Xi ) + F C[n] , ∀i. Thus D(X1 ) − a ∈ F C[n] , and for each i > 1 there exists Gi = Gi (X1 , . . . , Xi−1 ) ∈ C[X1 , . . . , Xi−1 ] such that D(Xi ) − Gi ∈ F C[n] . Since n X mi Ximi −1 D(Xi ) ∈ F C[n] and D(F ) = i=1 D(F ) = n X mi Ximi −1 (D(Xi ) − Gi ) + mi Ximi −1 Gi , i=1 i=1 where G1 = a, we obtain n X n X mi Ximi −1 Gi ∈ F C[n] and then obviously Gi = 0 i=1 for all i. Thus d is the zero derivation. Lemma 8 Let d be a linear derivation of Bnm and [aij ] its associated matrix. Then d is locally nilpotent if and only if [aij ] is nilpotent. Proof. The following equality can be verified by induction over s.  ds (x1 )   ..   . s d (xn )   x1  s = [aij ]  ...  . xn  (3.3) We know that d is locally nilpotent if and only if there exists r ∈ N such that dr (xi ) = 0 for all i. As {x1 , . . . , xn } is linearly independent over C by the (3.3), the result follows. ♦ Theorem 9 If d ∈ LN D(Bnm ) is linear and m = (m1 , . . . , mn ) wich mi ≥ 3, then d is the zero derivation. Proof. Since d is locally nilpotent, [d] is nilpotent (by Lemma 8) and diagonal (by Theorem 5). Thus, the matrix [d] is null and d is the zero derivation. ♦ In the case m = (2, . . . , 2), linear locally nilpotent derivations of the ring Bnm were characterized as follows. Theorem 10 [1, Theorem 1] If d ∈ Der(Bnm ) is linear and m = (2, . . . , 2), then d ∈ LN D(Bn2 ) if, and only if, its associated matrix is nilpotent and skewsymmetric. ♦ 7 4 Ring of constants In this section we show that the ring of constants of all nonzero linear derivations of Bnm , where m = (m1 , . . . , mn ) and mi ≥ 3, is trivial, that is ker(d) = C. During all this section we always consider m = (m1 , . . . , mn ) with mi ≥ 3. The next result ensures the existence of Darboux elements for every nonzero linear derivation of Bnm . Proposition 11 Let d be a nonzero linear derivation of Bnm . If d(xi ) = mαi xi , i = 1, . . . , n, for some α ∈ C∗ , then f = bxi11 · · · xinn is a Darboux element of d, d(f ) = λf , and   i1 i2 in λ=α . + + ···+ m1 m2 mn Proof. Let f = bxi11 · · · xinn . Then d(f ) =d(bxi11 · · · xinn ) =bd(xi11 · · · xinn ) n X =b ik xi11 · · · xikk −1 · · · xinn d(xk ) =b k=1 n X ik xi11 k=1 n X =αb k=1 =bα =λf  · · · xikk −1 · · · xinn  α xk mk  ik i1 x · · · xinn mk 1 i1 i2 in + + ··· + m1 m2 mn  xi11 · · · xinn ♦ Corollary 12 Let d be a nonzero linear derivation of Bnm . If f is a homogeneous element of degree k, then f is a Darboux element of Bnm with eigenvalue k λ= m . Proof. Let f = X a(i1 ···in ) xi11 · · · xinn be a homogeneous element of degree i1 +···+in =k k, where 0 ≤ in < m and a(i1 ···in ) ∈ C. 8 X d(f ) =d a(i1 ···in ) xi11 · · · xinn i1 +···+in =k = X a(i1 ···in ) d(xi11 · · · xinn ) X a(i1 ···in )  X a(i1 ···in ) k i1 x · · · xinn m 1 ! i1 +···+in =k = i1 +···+in =k = i1 +···+in =k k = m X i1 i2 in + + ···+ m m m a(i1 ···in ) xi11 i1 +···+in =k =λf · · · xinn  xi11 · · · xinn ! ♦ The main result this section is: Theorem 13 Let d be a nonzero linear derivation of Bnm . Then ker(d) = C. Proof. By Theorem 5, d(xi ) = α 6= 0. Let 0 6= f ∈ Bnm α mi xi for i = 1, . . . , n and α ∈ C. Since d 6= 0, so X b(i1 ,...,in ) xi11 · · · xinn such that d(f ) = 0. Thus f = (i1 ,...,in )∈I where 0 6= b(i1 ,...,in ) ∈ C for all (i1 , . . . , in ) ∈ I. Then X b(i1 ···in ) d(xi11 · · · xinn )   X i2 in i1 xi11 · · · xinn + + ···+ = b(i1 ···in ) α m1 m2 mn 0 = d(f ) = i1 i2 It follows from Lemma 1 that b(i1 ···in ) α( m + m + · · · + minn ) 6= 0 for all 1 2 (i1 , . . . , in ) ∈ I, because b(i1 ···in ) α 6= 0 for all (i1 , . . . , in ) ∈ I. So mi11 + mi22 + · · · + minn = 0 for all (i1 , . . . , in ) ∈ I. This implies that (i1 , . . . , in ) = (0, . . . , 0) for all (i1 , . . . , in ) ∈ I. Therefore f ∈ C. ♦ Theorem 14 Let d ∈ Der(Bnm ) be given by d(xi ) = mαi , i = 1, . . . , n, for some α ∈ C. If X a(i1 ,...,in ) xi11 · · · xinn f= (i1 ,...,in )∈I is a darboux element of d, this is, d(f ) = λf for some λ ∈ Bnm , then i1 λ = α( m + mi22 + · · · + minn ) for all (i1 , . . . , in ) ∈ I. 1 9 X Proof. Let f = a(i1 ,...,in ) xi11 · · · xinn . It follows from Theorems 13 and 5 that X a(i1 ···in ) b(i1 ...in ) xi11 · · · xinn d(f ) = (i1 ,...,in )∈I where b(i1 ···in ) = So X α( mi11 + i2 m2 + ··· + in mn ) ∈ C. Then a(i1 ···in ) b(i1 ...in ) xi11 · · · xinn = d(f ) = λf = X λa(i1 ···in ) xi11 · · · xinn a(i1 ...in ) b(i1 ···in ) = λa(i1 ...in ) for all (i1 , . . . , in ) ∈ I, by Lemma 1. Therefore λ = b(i1 ...in ) = α( mi11 + · · · + minn ) for all (i1 , . . . , in ) ∈ I. 5 i2 m2 + ♦ The case m = (2, . . . , 2) In this section we focus on the Fermat rings Bnm = C[X1 , . . . , Xn ] , (X12 + · · · + Xn2 ) where m = (2, . . . , 2) and n ≥ 3. For simplicity we denote Bnm by Bn2 . We study linear derivations of Bn2 , their rings of constants, and we show how to construct examples of linear derivations with trivial ring of constants. The next result will be useful for this purpose. Proposition 15 Let d be a nonzero linear derivation of Bnm , with d = dα + ds , where dα is the scalar derivation and is skew-symmetric derivation. Let dα definid by dα (xi ) = αxi for i = 1, . . . , n and α ∈ C. If f ∈ Bn2 is homogeneous element of degree k then d(f ) = λf if, and only if, ds (f ) = (λ − kα)f . Proof. It is easy to see that dα (f ) = kaf . Suppose that d(f ) = λf . Then λf = d(f ) = dα (f ) + ds (f ) = −αkf + ds (f ). Hence, ds (f ) = (λ − kα)f. Now suppose d1 (f ) = (λ − ka)f . Then d(f ) = dα (f ) + ds (f ) = kαf + (λ − kα)f = λf. ♦ Corollary 16 Let d = dα + ds be a nonzero linear derivation of Bn2 , where dα is the scalar derivation and ds is skew-symmetric derivation. If f ∈ Bn2 is homogeneous element of degree k then d(f ) = 0 if, and only if, ds (f ) = −kαf . 10 Proof. Consider λ = 0 in Proposition 15. ♦ The next Theorem shows that every skew-symmetric derivation has a nontrivial ring of constants and every nonzero scalar derivation has trivial ring of constants. Theorem 17 Let d = dα + ds be a nonzero linear derivation of Bn2 , where dα is the scalar derivation and ds is skew-symmetric derivation. Then 1. If ds is zero the derivation, then ker(d) = ker(dα ) is trivial. 2. If dα is the zero derivation, then ker(d) = ker(ds ) is nontrivial. Proof. 1) Observe that dα (f ) = kaf for all homogeneous element of degree k and dα (Vk ) ⊂ Vk . 2) It suffices to prove that there is f ∈ Bn2 such that ds (f ) = 0 and f 6∈ C. Let f a homogeneous element of degree 2 of Bn2 , then f = XAX T where A = [aij ] is a symmetric matrix and X = (x1 , . . . , xn ). Observe that for B = [ds ] we have d(f ) = d(XAX T ) =(XB)AX T + XA(XB)T =XBAX T + XA(−BX T ) =XBAX T − XABX T =X(BA − AB)X T . If B 2 6= 0 consider the symmetric matrix B 2 . It follows from the above remark that for f = XB 2 X T we have ds (f ) = 0 and f 6∈ C, because A = B 2 6= 0. If B 2 = 0, then λ = 0 is eigenvalue of B and B T . In this case, choose a nonzero element f = a1 x1 + · · · + an xn such that the nonzero vector (a1 , . . . , an )T is an eigenvector of B T . So d(f ) = 0 and f 6∈ C. Therefore, ker(ds ) 6= C. ♦ We also show that there are linear derivations with dα 6= 0, ds 6= 0, and trivial ring of constants. Theorem 18 Let ds be a nonzero skew-symmetric derivation of Bn2 . Then exists a scalar derivation dα of Bn2 such that the derivation d = dα + ds satisfies ker(d) = C. Proof. First note that the vector space Vk (the set of homogeneous elements of degree k of Bn2 ) is invariant with respect to ds . Hence ds (f ) = 0 if only if ds (fk ) = 0 for all homogeneous components fk of f . As a consequence of this fact we assume that f is a homogeneous element of degree k. Let α be a nonzero complex number that satisfies the conditions 1. α ∈ / Spec(ds ), 2. for all positive integer k, −kα ∈ / Spec(ds | Vk ). 11 This number exists because C is uncountable and the set of the numbers that satisfies 1) and 2) are countable. Let α be a number that satisfies the conditions 1) and 2), then ds (f ) 6= αf and ds (f ) 6= −kαf , for all f ∈ / C and for all positive integer k . Let dα be a scalar derivation defined by dα (xi ) = αxi , i = 1, . . . , n. Finally, by considering the derivation d = dα + ds , we show that ker(d) = C. In order to do that, if g ∈ Bn2 is a nonzero homogeneous element of degree k, then d(g) = 0 if, and only if, ds (g) = −kαg, by Corollary 16, which implies k = 0. Therefore, g ∈ C∗ . ♦ We now provide and explicit example of such derivation: Example 19 Let d = d1 + ds be the linear derivation of B32 = C[x, y, z] given by       1 0 0 1 0 0 0 0 0 [d] = [d1 ] + [ds ] =  0 1 −1  =  0 1 0  +  0 0 −1  . 0 1 1 0 0 1 0 1 0 We claim that ker(d) = C. To be more precise, let Vk be the set of all homogeneous elements of degree k in C[y, z]. The set Sk = {y k , y k−1 z, . . . , yz k−1 , z k } is a basis for Vk . The matrix  k 1 0  −k k 2    0 −(k − 1) k   0 −(k − 2) [d | Vk ] =    . . . .  . . 0   ..  0 0 . 0 0 0 ... ... .. . ... ... 0 0 .. . .. . k−2 0 .. . k k−1 −2 0 k −1 ... 0 0 .. .        0    0    k  k is the matrix the linear derivation d restrict to subspace Vk in the basis Sk . It is easy check that Det([d | Vk ]) 6= 0 for all k ≥ 1, by the principle of induction. Then d(f ) 6= 0 for all homogeneous elements of degree k ≥ 1. Therefore, ker(d) = C. Theorem 20 Let d = dα + ds be a nonzero linear derivation of Bn2 , where dα is a scalar derivation and ds is a skew-symmetric derivation. If ds is a locally nilpotent derivation then ker(d) = C, for all nonzero scalar derivation dα . Proof. Let 0 6= f ∈ Bn2 such that d(f ) = 0. It suffices to show that f ∈ C. We may assume that f is a nonzero homogeneous element of degree k, because Vk is invariant by d. Let m be the smallest positive integer such that g = dm−1 (f ) 6= 0 s and dm (f ) = 0, this m exists because d is a locally nilpotent derivation. This s s 12 implies that g is a nonzero homogeneous element of degree k, because Vk is invariant by d. One easily verifies that dα ds = ds dα and dds = ds d. Note that d(g) = dα (g) + ds (g) = kαg, because g is an homogeneous element of degree k and dm s (f ) = 0. Now observe that (f )) = dm−1 (d(f )) = dm−1 (0) = 0. d(g) = d(dm−1 s s s We thus get kαg = 0. And so k = 0. Therefore, f ∈ C. ♦ To conclude this section we construct two families of examples which illustrate this theorem. Example 21 Let n ≥ 3 be an odd number defined by the skew-symmetric matrix n × n  0 0 ... 0  0 0 ... 0   .. .. . . .  . .. [ds ] =  . .  0 0 ... 0   0 0 ... 0 1 i ... 1 and ds a linear derivation of Bn2 0 0 .. . 0 0 i −1 −i .. .      . −1   −i  0 It is easy to check that [ds ]3 = 0, which implies that [ds ] is nilpotent. Then ds is a locally nilpotent linear derivation of Bn2 , by Theorem 10. Now consider the linear derivation d = d1 + ds , where d1 (xi ) = xi for i = 1, . . . , n. It follows from Theorem 20 that ker(ds ) = C. Example 22 Let n ≥ 4 be an even number and ε ∈ C a primitive (n − 1)-th root of unity. Set ds a linear derivation of Bn2 by the skew-symmetric matrix n × n   0 0 ... 0 ... 0 −1  0 0 0 ... 0 0 −ε     .. .. . .  .. . . .. ..  . .  . . . . .   k  0 0 . . . 0 . . . 0 −ε [ds ] =     . . .  . . . . . . .. .. .. ..  .. ..    n−2   0 0 ... 0 ... 0 −ε 1 ε . . . εk . . . εn−2 0 Again, [ds ] is nilpotent ([de ]3 = 0). Thus, ds is a locally nilpotent derivation of Bn2 , by Theorem 10. Now considering the linear derivation d = d1 + de , where d1 is the same as in the previous example, we conclude that ker(ds ) = C, by Theorem 20. Remark: In the Example 19 it easy see that [ds ] is not nilpotent and, consequently ds is not locally nilpotent (Theorem 10). This shows that ds locally nilpotent is not a necessary condition in Theorem 20 for ker(dα + ds ) = C. 13 References [1] P. Brumatti and M. Veloso, On locally nilpotent derivations of Fermat Rings, Algebra and Discrete Mathematics 16 (1), 33–41, (2013). [2] P. Brumatti and M. Veloso, A note on Nakai’s conjecture for the ring K[X1 ,...,Xn ] , Colloquium Mathematicum 123 (2), 277–283, (2011). m (a X 1 +···+a X mn ) 1 1 n n [3] D. Fiston and S. Maubach, Constructing (almost) rigid rings and a UFD having infinitely generated Derksen and Makar-Limanov invariant, Canadian Mathematical Bulletin 53 (1), 77-86, (2010). [4] G. Freudenberg, Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia of Mathematical Sciences 136, Springer-Verlag Berlin Heidelberg, (2006). [5] A. Nowicki, On the nonexistence of rational first integrals for systems of linear differential equations, Linear Algebra and Its Applications 235, 107120, (1996). [6] J. M. Ollagnie and A. Nowicki, Derivations of polynomial algebras without Darboux polynomials, Journal of Pure and Applied Algebra 212, 1626-1631, (2008). [7] J. Zielinski, Rational Constants of Generic LV Derivations and of Monomial Derivations, Bulletin of the Polish Academy of Sciences. Mathematics 61, Issue: 3, 201-208, (2013). 14