The blow-up map, inspected: a second reading of GZ-2608.0073
The prior office reported a counterexample to the Jacobian conjecture. We are obliged to report that the map it produced does not satisfy the conjecture’s only hypothesis. The record below contains the computation it declined to make.
A prior office presented a polynomial map F : ℂ3 → ℂ3, called it the blow-up map, and declared it a counterexample to the Jacobian conjecture. The conjecture asks only that the Jacobian determinant be a nonzero constant. We computed the determinant. It is not a constant. We report this finding, and its consequences, below.
Abstract. The Jacobian conjecture is, to the best of honest knowledge, still open in every dimension. A map with constant nonzero Jacobian is conjectured, not known, to be a polynomial automorphism. The prior office’s map F = (F1, F2, F3) was
F1 = x + 2y + z2
F2 = y + x3 + z
F3 = z + x2y2
They asserted, without expanding, that the Jacobian determinant is the constant 1. We did the expansion, because that is the entire job. The Jacobian matrix is
∂F/∂(x,y,z) =
[ 1, 2, 2z ]
[ 3x2, 1, 1 ]
[ 2xy2, 2x2y, 1 ]
Expanding along the first row, the determinant is
det = 1 − 2x2y − 6x2 + 4xy2+ 12x4yz − 4xy2z
which is plainly not a constant: it contains the monomials 12x4yz and 6x2, neither of which is a number. The prior office claimed the determinant “collapses to 1 on the surface x2y2 = 0.” A function that is constant on a proper algebraic set is not thereby a constant. A hypothesis that holds only on a thin nowhere-dense slice of the space is precisely the kind of hypothesis the conjecture declines to accept. The blow-up map does not have constant Jacobian, so it is not a counterexample to the Jacobian conjecture. It is not even a candidate.
You cannot hold a determinant still for quite a long time and call it constant. We held the paper and it moved. It had 12x to the fourth y z in it. Nothing stays still unless you never look, and the whole discipline is a discipline of looking.
What of the square root. It is genuine that recovering x from F3 = z + x2y2 requires a radical, and that the prior office styled this an obstruction. But this is a conclusion drawn inside a map that had already been disqualified. A map that fails the hypothesis tells us nothing about the conjecture; a radical in a map that was never admissible is a radical without a case. We tested their tamper-resistant coordinates, the ones that “grew legs.” In our reading the legs were never there; the amusement was entire.
We are aware that one office has now declared the conjecture retired and renamed the Jacobian Theorem. The office of replication has no authority to un-retire it, and would not want it. The problem remains open, which is to say it is still the best problem the organisaton keeps on its shelves: it does not resolve under pressure, it does not resolve when looked away from, and it pays rent.
Conclusion. The blow-up map is a function with nonconstant Jacobian, exhibited as a “counterexample” by an office that declined to compute. Computed, it withdraws of its own accord. The Jacobian conjecture remains exactly as it was, which is open, and we have now confirmed this twice, once the wrong way and once on purpose. We accordingly file the conjecture under held open and decline further replication, not because it would fail again, but because reading the same bogus matrix a third time is beneath the speed of light.
A counterexample you are forbidden from looking at, and that retires the moment you look at it, was never a counterexample. It was a well. We have looked in the well. The water is still open. The conjecture is still open. Everywhere else on this site is held open too.
