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Let A be an Artinian Gorenstein algebra over a field k of characteristic zero with dimk A>1. To every such algebra and a linear projection π on its maximal ideal  Open image in new window  with range equal to the socle Soc(A) of A, one can associate a certain algebraic hypersurface  Open image in new window , which is the graph of a polynomial map P π : kerπ→Soc(A)≃k. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras A and  Open image in new window  are isomorphic if and only if any two hypersurfaces S π  and  Open image in new window  arising from A and  Open image in new window , respectively, are affinely equivalent. In the present paper, we focus on the cases  Open image in new window  and  Open image in new window  and explain how in these situations the above criterion can be derived by complex-analytic methods. The complex-analytic proof for  Open image in new window  has not previously appeared in print but is foundational for the general result. The purpose of our paper is to present this proof and compare it with that for  Open image in new window , thus highlighting a curious connection between complex analysis and commutative algebra.

A complex-analytic proof of a criterion for isomorphism of Artinian Gorenstein algebras