Journal of the Ramanujan Mathematical Society

Volume 28A, Issue SPL, July – Special Issue 2013 pp. 221–246.

Surjective derivations in small dimensions

Authors: R. V. Gurjar, K. Masuda and M. Miyanishi
Author institution: Research Center for Mathematical Sciences, Kwansei Gakuin University, 2-1 Gakuen, Sanda, 669-1337, Japan

Summary: Let D be a C-derivation on a polynomial ring C[x1, x2]. Cerveau [4] asserts that D is surjective as a linear mapping if and only if D = ∂/∂ x1 + ax2 ∂/∂ x2 with respect to a suitable algebraic change of coordinates of C[x1, x2], where a ∈ C. Inspired by various results in [4], we consider a surjective derivation defined on an affine domain over C of dimension one or two. Though our proofs are mostly algebraic or algebro-geometric, the idea using a result of Dimca-Saito [5] which is behind the arguments in [4] and based on the differential complex of the polynomial ring C [x1, …, xn] is inspiring and affects our arguments.

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