A companion to Terence Tao’s digestion of the Alpöge–Fable counterexample to the (complex) Jacobian conjecture. The construction lives on a 3-dimensional variety X = {(a,b,c,d,e) : Res(L,Q)=1, ad+bc=1}, which is a copy of ℝ³ carrying two coordinate systems. Here L(s,t)=a·s+b·t is a linear form and Q(s,t)=c·s²+d·s·t+e·t² a quadratic one; the map of interest is multiplication F(L,Q)=L·Q, a cubic. Edit either coordinate system below — by typing or with the arrows — and watch all the data, the plot of L, Q and L·Q, and the roots update. When the cubic has three real roots the multiplication map is 3-to-1: click a root of Q on the plot to re-factor the same cubic and land on a different point of X — the map’s failure of injectivity, made visible.
Companion to the blog post “A digestion of the Jacobian conjecture counterexample”, whose underlying mathematics Tao worked out in an extended conversation with an AI chatbot. Runs entirely in your browser.
The master chart — a global copy of ℝ³, valid everywhere.
Editing a here holds y, z fixed. Arrows step by 0.1.
A derived chart, singular at a=0.
Editing a here holds b, c fixed — a different motion. Setting a=0 snaps b=c=1 and keeps d, e.
Curves are graphs over x = s/t (the chart t=1); dots on the axis are real roots (filled = L’s, open = Q’s). When Q’s roots are complex the plane doubles as the s/t-plane and they appear as a faded pair above and below the axis (height = imaginary part).
| d, e (derived) | , |
|---|---|
| L(s,t) | |
| Q(s,t) | |
| F(L,Q) = L·Q | |
| Res(L,Q) | |
| ad + bc | |
| root of L | |
| roots of Q |
The point of X corresponds to the original variables (z₁, z₂, z₃) = (a, y, −2z), at which the map of equation (1) evaluates to F(z₁,z₂,z₃) = (be, 2(ae+bd), 2ac) — a linear reindexing of the coefficients of L·Q. So the original map is the multiplication map in disguise (constant Jacobian −2), and the three Collision presets are its three coinciding preimages — the three factorizations of one cubic.
| (z₁, z₂, z₃) | |
|---|---|
| F(z₁, z₂, z₃) |