On the Jacobian Question

Abstract

The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic approach is shown to be extendable to higher dimensions by proceeding on exactly identical lines. As per the important result due to Bass, Connell, and Wright [3] it is enough to show the validity of the Jacobian conjecture for cubic polynomials of special type (BCW form) in two, three, and n variables. Firstly, the method of proof for the cases of two and three variables is discussed at length. It is then indicated that the extension to the several variables case follows automatically by just following the same steps and there is no hindrance as there is essentially no change in the basic situation and the same line of thought used for the case of two and three variables remains applicable. Thus, we show that the problem can be solved completely using the important reduction of the problem to the case of special cubic degree polynomials [3]. We have shown how to obtain inverse polynomials. We fully obtain them for two variables case and almost obtain them for three variables case.