Algebraic observations on the Jacobian conjecture

AbstractThis paper contains conditions that are equivalent to the Jacobian Conjecture (JC) in two variables and partial results toward establishing these conditions. If u and v are a Jacobian pair of polynomials in k[x,y] which provide a counterexample then by a change of variables there is a Jacobian pair which generate an ideal of the form 〈p(x),y〉. (A similar result holds for an arbitrary number of variables.) JC follows if p(x) must be linear or equivalently if p′(x) is constant. Conditions which yield this result are derived from the Jacobian relation and the fact that 〈u,v〉=〈p(x),y〉. Other conditions that imply JC are derived from the fact that JC follows if the ring k[x,u,v]=k[x,y] when the Jacobian determinant of u and v is 1. One easily arranges for k[x,y] to be integral over k[x,u,v] with the same quotient fields. For example, if k[x,u,v] must be square-root closed in k[x,y], JC follows. The paper studies the case where j(u,v)=1 forces k[x,u,v] to be seminormal in k[x,y]. In this case and in other cases, y satisfies a monic polynomial of degree two over a localization of k[x,u,v]. Conditions that imply k[x,u,v]=k[x,y] include equality after taking quotient rings and localizing. Other conditions which imply equality involve showing k[x,u,v] is a regular ring and computation of the conductor ideal which must be principal as an ideal of k[x,y]