Abstract
The classical Jacobian Conjecture asserts that every locally invertible polynomial self-map of the complex affine space is globally invertible. A Keller map is a (hypothetical) counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a map between the Picard groups of suitable compactifications of the affine plane, that satisfy a complicated set of conditions. This is essentially a combinatorial problem. Several solutions to it ("frameworks") are described in detail. Each framework corresponds to a large system of equations, whose solution would lead to a Keller map.
Highlights
Suppose f (x, y) and g(x, y) are two polynomials with complex coefficients
The map is ramified above three points: intersection with the (-2)-curve, (-3)-curve, and (-4)-curve, that we will identify with {0}, {∞}, and {1} respectively
They correspond to the branches that end with the 0-curve with the self-intersection (-2), with the 0-curve with the self-intersection (-1), and with the forked (-5)-curve
Summary
Suppose f (x, y) and g(x, y) are two polynomials with complex coefficients. The classical Jacobian Conjecture (due to Keller, [10]) asserts the following. Suppose Ei and Ej are two intersecting curves at infinity This means that on the graph of curves, we have an edge connecting Ei and Ej. We will call a pair of rational functions on our surface (fi, fj) a local coordinate system for that edge if (fi)|Ei. Note that the coordinate system of an edge is far from unique, and that fi may have other zeros and poles, possibly intersecting Ej, and the same for fj and Ei. On the other hand, a local coordinate system exists for every edge of our graph of curves, which can be proven by induction. There does not seem to be a way to modify it to get a Keller map, even by introducing additional variables
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