The 2-dimensional Jacobian Conjecture: A Computational Approach

Summary

The Jacobian Conjecture is one of the most important open problems in algebraic geometry. It was included among the millennium open problems in mathematics by Steve Smale in his address to the Millennium ICM (in the year 2000). Given a polynomial mapping F: ℂⁿ→ℂⁿ which has a nonzero constant determinant of its Jacobian matrix (The Jacobian Condition), the conjecture is that F is an invertible morphism. This means that it is injective, surjective and that its inverse mapping F ⁻¹ is also polynomial. Even in dimension n=2 this is still open. This article is about the 2-dimensional Jacobian Conjecture. The degree of a polynomial mapping F is the maximum degree of its polynomial coordinate functions. A striking fact that we prove is that given a degree d, the 2-dimensional Jacobian Conjecture can be settled for all the polynomial mappings of degree d or less. If it is disproved then a counterexample is constructed. The magic tool used is the machinery of Gröbner bases. We use the powerful tools of computational algebra and in particular the theory of Gröbner bases in order to solve a certain ideal membership problem in an algebra of many variables polynomials over the complex field. The Jacobian condition satisfied by the mapping F (of degree d or less) is used in a canonical way to construct an ideal (the Jacobian ideal) within this algebra of polynomials. We consider the two relative resultants of the mapping F (one with respect to X and the second with respect to Y). The key theorem we prove is that the Jacobian Conjecture is valid for F if and only if the leading coefficients of these two resultants belong to the Jacobian ideal. We call this result: The resultant reformulation of the Jacobian Conjecture. Now the Gröbner bases machinery comes in to help to decide on the ideal membership problem we have. The algorithm was programmed and was used to prove the 2-dimensional Jacobian Conjecture up to degree 15. The theoretical importance of this algorithm is that it shows that the conjecture is a decidable problem.

The article contains many other important results, both new and old that are relevant to this particular computational approach to this famous open problem. We assume that the reader has some background in the theory and the folklore of the Jacobian Conjecture. The classical paper [H. Bass, E. Connell, and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (2), 7 (1982), 287–330] and the only comprehensive book in this area [A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, Vol. 190, Birkhäuser, Basel, 2000] are excellent sources. We use the notation introduced in those sources whenever possible.