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By Mr (i: R (n and Daigle Daniel (-ottw-ms

Abstract

On polynomials in three variables annihilated by two locally nilpotent derivations. (English summary) J. Algebra 310 (2007), no. 1, 303–324. The main result of the article under review asserts that in a polynomial ring B in three variables over an algebraically closed field k of characteristic zero, a nonconstant irreducible polynomial f is annihilated by two locally nilpotent derivations with distinct kernels if and only if B ⊗k[f] k(f) is isomorphic to a k(f)-algebra of the form k(f)[X, Y, Z]/(XY − ϕ(Z)) for some nonconstant polynomial ϕ(Z) ∈ k(f)[Z]. In a geometric setting, this means equivalently that a nonconstant regular function f: A3 → A1 on the affine three space is invariant under two actions of the additive group Ga,k with distinct general orbits if and only if its generic fiber is isomorphic to a surface in A3 k(f) defined by an equation of the form XY = ϕ(Z). This characterization is strongly related to the problem of finding which smooth affine surfaces S equipped with two Ga,k-actions with distinct general orbits (one says equivalently that S has a trivial Makar-Limanov invariant) can be realized, possibly in a nonequivariant way, as hypersurface

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.4184
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