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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

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