{\bf Invariants of algebraic varieties}

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\centerline{Vladimir Shpilrain, City University of New York}


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This is a recent joint work with Jie-Tai Yu.
We address the following two general
problems: given two algebraic varieties in $C^n$, find out whether or not
they are (1) isomorphic; (2) equivalent under an automorphism of $C^n$.
Although a complete solution of either of those problems is out of the
question at this time, we give some handy and useful invariants of
isomorphic as well as of equivalent varieties. Furthermore, and more
importantly, we give a universal procedure for obtaining all possible
algebraic varieties isomorphic to a given one, and use it to construct
numerous examples of hypersurfaces with non-unique embeddings in $C^n$,
with a  view to a counterexample to the  Embedding conjecture of Abhyankar
and  Sathaye.
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