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Jacobian Conjecture as a Problem on Integral Points on Affine Curves


It is shown that the Jacobian conjecture over number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over \(\mathbb {C}\) is false, then for some n ≫ 1 there exists a counterexample \(F\in \mathbb {Z}[X]^{n}\) of the form \(F_{i}(X)=X_{i}+ (a_{i1}X_{1}+\cdots +a_{in}X_{n})^{d_{i}}\), \(a_{ij}\in \mathbb {Z}\), di = 2;3, \(i,j=1,\dots ,n\), such that the affine curve F1(X) = F2(X) = ⋯ = Fn(X) has no non-zero integer points.

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The author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.04-2019.316, and Vietnam Institute for Advanced Study in Mathematics (VIASM).

The author wishes to express his thank to the professors Arno van den Essen, Ha Huy Vui and Ludwick. M. Drużkowski for some valuable discussions. The author would like to thank the Vietnam Institute for Advanced Study in Mathematics for their helps.

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Correspondence to Chau Van Nguyen.

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Van Nguyen, C. Jacobian Conjecture as a Problem on Integral Points on Affine Curves. Vietnam J. Math. 50, 195–204 (2022).

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