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The Nagata automorphism is wild.
Ivan P Shestakov1, Ualbai U Umirbaev
1Instituto de Matemática e Estatistica, Universidade de São Paulo, Caixa Postal 66281, São Paulo, SP 05311-970, Brazil. shestak@ime.usp.br
Summary
The Nagata automorphism in polynomial rings is proven to be wild. This means it cannot be broken down into simpler elementary automorphisms, a key finding in algebraic geometry.
Area of Science:
- Algebraic Geometry
- Commutative Algebra
Background:
- The Nagata automorphism is a significant example in the study of polynomial rings.
- Understanding the structure of automorphisms is crucial for classifying algebraic varieties.
Purpose of the Study:
- To determine the nature of the Nagata automorphism.
- To classify the Nagata automorphism as either tame or wild.
Main Methods:
- The study involves advanced techniques in the theory of polynomial rings.
- Analysis of the automorphism's decomposability into elementary components.
Main Results:
- The Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is definitively proven to be wild.
- This wild nature implies it cannot be expressed as a composition of elementary automorphisms.
Conclusions:
- The Nagata automorphism is a wild automorphism.
- This classification has implications for understanding the structure and classification of automorphisms in polynomial rings.