Very degenerate polynomial submersions and counterexamples to the real Jacobian conjecture |
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Institution: | 1. Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos, São Paulo, Brazil;2. Departamento de Matemática, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo 13566-590 São Carlos, São Paulo, Brazil |
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Abstract: | Until recently, the only known examples of non-injective polynomial local diffeomorphisms of the plane were the Pinchuk maps discovered in 1994. These maps have the form , where p is a fixed polynomial of degree 10, and q satisfies certain relation. The lowest possible degree of q in a Pinchuk map is 25. In 2021, with the same p of a Pinchuk map, another example was given with q of degree 15. Aiming to find different examples, with the first components having lower degrees, we look for polynomials of degree less than or equal to 9 sharing some properties with the polynomial p of a Pinchuk map. We find precisely one polynomial of degree 7 and some ones of degree 9. By using one of the polynomials of degree 9 in the first component we construct a non-injective polynomial local diffeomorphism of the plane, with degree 15. |
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Keywords: | Polynomial submersions Global injectivity Pinchuk maps |
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