Note on boundary obstruction to Jacobian conjecture of two variables (Q2479923)

Let \((f,g)\) be a polynomial mapping from \(\mathbb{C}^2\) to \(\mathbb{C}^2\), with \(f,g\in \mathbb{C}[x,y]\); let \(J(f,g)\) denote the Jacobian determinant \(f_{x} g_{y} - f_ {y} g_{x}\). The Jacobian conjecture states that if \(J(f,g)\) is a non-zero constant, then \((f,g)\) has an inverse polynomial mapping. In the paper [Kodai Math. J. 6, 419--433 (1983; Zbl 0526.13013)], the author studied this conjecture following the methods described by [\textit{S. S. Abhyankar}, Lectures on expansion techniques in algebraic geometry. With notes by Balwant Singh. Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research. 57. (Bombay): Tata Institute of Fundamental Research, (1977; Zbl 0818.14001)], i.e. expansions of polynomials as sums of weighted homogeneous polynomials and the use of Newton polyhedrons. The present note is a rewritten version of that paper, with some new result and several new examples: as a strategy to prove (or disprove) the Jacobian conjecture in two variables, the author suggests to show that there is no polynomial which satisfies certain four necessary conditions expressed in the Main Theorem.