COLLOQUIUM MATHEMATICUM VOL. 123 2011 NO. 2 A NOTE ON NAKAI’S CONJECTURE FOR THE RING K[X 1 ,...,X n ]/(a 1 X m 1 + ··· + a n X m n ) BY PAULO ROBERTO BRUMATTI (Campinas) and MARCELO OLIVEIRA VELOSO (Ouro Branco) Abstract. Let k be a field of characteristic zero, k[X1,...,Xn] the polynomial ring, and B the ring k[X1,...,Xn]/(a1X m 1 + ··· + amX m n ), 0 = ai ∈ k for all i and m, n ∈ N with n ≥ 2 and m ≥ 1. Let Der 2 k (B) be the B-module of all second order k-derivations of B and der 2 k (B) = Der 1 k (B) + Der 1 k (B) Der 1 k (B) where Der 1 k (B) is the B-module of k- derivations of B. If m ≥ 2 we exhibit explicitly a second order derivation D ∈ Der 2 k (B) such that D/ ∈ der 2 k (B) and thus we prove that Nakai’s conjecture is true for the k-algebra B. Introduction. Throughout this paper k denotes a field of characteristic zero. Let S = k[X 1 ,...,X n ] be the polynomial ring in n variables over a field k and let A = S/J be an affine k-algebra. Let Der n k (A) be the A-module of k-derivations of order ≤ n where 1 ≤ n ∈ N. Let Der k (A) be the k- algebra  n Der n k (A) and der k (A) the subalgebra generated by Der 1 k (A). The set der k (A) ∩ Der n k (A) will be denoted by der n k (A). In [1], Grothendieck has shown that Der k (A) = der k (A) if A is regular. The Nakai conjecture states the converse. In 1986 Singh [5] presented the following conjecture, which is stronger than Nakai’s conjecture: If A = S/(F ) and Der 2 k (A) = der 2 k (A) then A is regular. Singh’s conjecture for a generic affine k-algebra A is not valid. A coun- terexample can be found in [4]. In this work we prove that Singh’s conjecture is true in the following cases: (1) B = S/(F ), where F = a 1 X m 1 + ··· + a m X m n with 0 = a i ∈ k for all i (Theorem 6). (2) C = S/(H ) where H ∈ S is homogeneous of degree ≤ 2 (Corollary 7). 1. A set of generators for Der 1 k (B). Let B be a ring S/(F ), where F = a 1 X m 1 + ··· + a m X m n with 0 = a i ∈ k. In this section we give a set of generators for the B-module Der 1 k (B). 2010 Mathematics Subject Classification : 13N05, 13B10, 13N10, 13N15. Key words and phrases : Nakai’s conjecture, derivations, commutative algebra. DOI: 10.4064/cm123-2-10 [277] c  Instytut Matematyczny PAN, 2011