lectures on Jacobian conjecture
Ian Agol
Date: Friday, November 5, 2004 10:52 AM -0500
From: Mel Hochster <hoch...@umich.edu>
To: math-s...@umich.edu
Cc: Craig Huneke <hun...@math.ukans.edu>, Irena Swanson
<iswa...@nmsu.edu>
Subject: The Jacobian conjecture in the plane is solved
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COMING ATTRACTION
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Special Seminar Series
The Jacobian conjecture in the plane has been an open problem
since 1939 (Keller). The simple statement is this: given a ring
map F of C[x,y] (the polynomial ring in two variables over the
complex numbers C) to itself that fixes C and sends x, y
to f, g, respectively, F is an automorphism if and only
if the Jacobian determinant f_x g_y - f_y g_x is a nonzero
element of C. The condition is easliy seen to be necessary.
Sufficiency is the challenge.
Carolyn Dean has proved the conjecture and will give a series
of talks on it beginning Thursday, December 2, 3-4 pm,
continuing on December 9 and December 16. Because there have
been at least five published incorrect proofs and innumerable
incorrect attempts, any announcement of a proof tends to
be received with skepticism. I have spent approximately
one hundred hours (beginning in mid-August) checking every
detail of the argument. It is correct.
Earlier papers established the following: if there is
a counterexample, the leading forms of f and g may
be assumed to have the form (x^a y^b)^J and (x^a y^b)^K,
where a and b are relatively prime and neither J
nor K divides the other (Abhyankar, 1977). It is known that
a and b cannot both be 1 (Lang, 1991) and that one may
assume that C[f,g] does not contain a degree one polynomial
in x, y (Formanek, 1994).
Let D_x and D_y indicate partial differentiation with respect
to x and y, respectively. A difficult result of Bass (1989)
asserts that if D is a non-zero operator that is a polynomial
over C in x D_x and y D_y, G is in C[x,y] and D(G)
is in C[f,g], then G is in C[f,g].
The proof proceeds by starting with f and g that give
a counterexample, and recursively constructing sequences of
elements and derivations with remarkable, intricate and
surprising relationships. Ultimately, a contradiction is
obtained by studying a sequence of positive integers associated
with the degrees of the elements constructed. One delicate
argument shows that the sequence is bounded. Another delicate
argument shows that it is not. Assuming the results described
above, the proof, while complicated, is remarkably self-contained
and can be understood with minimal background in algebra.
- Mel Hochster