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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
##### Keywords:
polynomial map; Jacobian conjecture; Newton-Puiseux chart
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