While testing problem 5 for
Homework 1,
I was discussed with GPT 5.6 first how to start
searching for counter examples to the still open
Jacobian conjecture in dimension 2. The expert system of
course knows that we need to search among polynomials of degree
125 or higher. It is clear that a brute force search is
unrealistic. I still insisted that I want to build a casino
which allows a user to play the lottery. The model indeed
had no problem to generate such a lab and even built a version
which felt a bit more like a slot machine.
casino.m
Save it import it to Mathematica.
You could also run from the command line like
"nohup wolfram < casino.m > out" and let it run for a few days.
The search actually already is a bit more sophisticated. The space
of polynomials of maximal degree n has dimension (n+1)(n+2)/2, so
that the linear space of polynomial maps F(x,y) = (p(x,y),q(x,y))
of maximal degree n has dimension (n+1)(n+2). The determinant
det(dF) is a polynomial of maximal degree 2n-2. The number of coefficients
which have to be zero is n(2n-1). Asking det(dF)=1 produces a system of 2n^2-n
equations. We have essentially twice as many equations than variables.
In order to have a chance of having a constant determinant already asks to
restrict the search to special families of polynomials.
It does not look so bad at first as for n=2, we have 12 coefficients and
6 equations. But for n=5 already we have 42 unknown coefficients and 45 equations
and for n=10, we have 132 unknown coefficients and 190 equations.
Ordinary dimension count tells nothing.
But then the problem only begins. Even if we find families of pairs (p,q)
which have constant determinant like maps of the form F(x,y) = (x,y+p(x))(x+q(y),y).
I hope you had fun exploring a bit the remaining open question about the Jacobian
conjecture. An other juicy question is what happens over a finite field F_p?
But the map F(x,y) = (x-xp,y) has det(dF) = 1 even so F is not an
automorphism on F_p x F_p. The Jacobian conjecture is false for any field of
positive characteristic. For characteristic 0, the 2-dimensional case is
the frontier. It is hard to estimate these days, when this problem will be solved.
If there is a counter example, it might take some months. If not, it might take
longer and would at the moment be hard to bet who wins, humans or machines.
