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An application of Newton-Puiseux charts to the Jacobian problem. (English) Zbl 1174.14054
The author studies 2-dimensional Jacobian maps \((f,g):\mathbb{C} ^{2}\rightarrow \mathbb{C}^{2}\) (i.e. \(f,g\) are polynomials and \(\text{Jac }(f,g)\equiv 1)\) using so-called Newton-Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The mapping \((f,g)\) expressed in these charts takes a very simple form, which allows him to obtain generalizations of the known theorem. As applications he proves that the Jacobian conjecture holds true for \((f,g)\) in the following cases:
1.
topological degree of \((f,g)\) is \(\leq 5\) (a generalization of the Domrina-Orevkov result),
2.
\(\text{GCD}(\deg f,\deg g)\leq 16\) (a generalization of the Heitmann result),
3.
\(\text{GCD}(\deg f,\deg g)\) is equal to 2 times a prime (a generalization of the Appelgate-Onishi result).

MSC:
14R15 Jacobian problem
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