On the shape of possible counterexamples to the Jacobian conjecture (Q342828)
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scientific article; zbMATH DE number 6654556
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the shape of possible counterexamples to the Jacobian conjecture |
scientific article; zbMATH DE number 6654556 |
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On the shape of possible counterexamples to the Jacobian conjecture (English)
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18 November 2016
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Jacobian conjecture
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counterexample
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In this long paper, the authors analyse (in the spirit of Abhyankar) the structure of Jacobian pairs \((P,Q)\) i.e. polynomials in two variables over a field of characteristic zero satisfying the Jacobian problem condition \( \text{Jac}(P,Q)=\text{const}\neq 0.\) There are many technical results concerning this structure. Results which may be easily presented are: If \((P,Q)\) is a counterexample to the Jacobian conjecture then:NEWLINENEWLINE1. \(\text{GCD}(\deg P,\deg Q)\geq 16\) (proved earlier by \textit{R. C. Heitmann} [J. Pure Appl. Algebra 64, No. 1, 35--72 (1990; Zbl 0704.13010)],NEWLINENEWLINE2. \(\text{GCD}(\deg P,\deg Q)\neq 2p\) for all primes \(p.\)
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