A pair days in the past, Levent Alpöge, a mathematician working at Anthropic, found a counterexample to the Jacobian conjecture utilizing Claude Fable 5.
I used to be curious whether or not most mathematicians have been making an attempt to show or disprove the conjecture, so I requested Claude.
Earlier than a counterexample to the Jacobian conjecture was discovered, did most mathematicians imagine it was true or false?
Claude’s response was
The premise of this query isn’t fairly proper — no counterexample to the Jacobian conjecture has been discovered. It stays an open downside in arithmetic: nobody has confirmed it true, and nobody has discovered a counterexample disproving it. … If you happen to encountered a declare {that a} counterexample was discovered, do you might have a supply for that? I’d be comfortable to look into it, since that may really be a serious lead to algebraic geometry if true.
After all Claude doesn’t know that it solved the conjecture. It didn’t even clear up the conjecture. It was an inanimate software within the hand of a mathematician, similar to a chunk of chalk or a dry erase marker.
The center a part of Claude’s response was that mathematicians are (have been) divided on whether or not the conjecture is true. So it was not just like the Riemann speculation, which most individuals imagine to be true, or the P = NP conjecture, which most individuals imagine to be false.
Now what’s the Jacobian conjecture? It says {that a} polynomial operate from ℝn to ℝn with fixed, non-zero Jacobian has a polynomial inverse. (The conjecture was said extra usually for fields of attribute 0, wherein the derivatives defining the Jacobian must be outlined algebraically, not when it comes to limits.)
Alpöge got here up with a counterexample, a polynomial operate from ℝ³ to ℝ³ with fixed Jacobian determinant −2. The operate is
It’s a tedious however easy calculus train to indicate that the determinant equals −2 in all places. The inverse operate theorem says {that a} operate is domestically invertible at any level the place the Jacobian determinant is non-zero, so Alpöge’s operate is domestically invertible in all places.
Nonetheless, the operate takes on some values greater than as soon as. For instance, (0, 0, −1/4) and (1, −3/2, 13/2) each map to (−1/4, 0, 0). Due to this fact the operate shouldn’t be invertible globally. So not solely does the operate not have a polynomial inverse, it doesn’t have an inverse even if you happen to enable non-polynomial features.
Alpöge’s counterexample disproves the Jacobian conjecture for n = 3. It will possibly trivially be prolonged to all n > 3 by defining the operate to be Alpöge’s operate for 3 variables and the identification for the remainder. The conjecture stays open for n = 2.
