> and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search

Is that true? Even restricting this to f(x,y,z) and coefficients and powers to 1 ≤ x ≤ 10, there are a lot of polynomials to check, and checking requires checking the Jacobian determinant and, if it’s a non zero constant, finding two points for which the polynomial produces the same value.

Or is there a way to generate all polynomials with a non-zero Jacobian determinant, and does that speed up things? (My intuition say it wouldn’t, because I guess those with zero determinants are rare)