I1 n=3 brackets of coordinates: a*x*y | b*y*z | c*x*z
I1 n=3 Jacobiator(x,y,z) = 0
I1 n=3 Jacobiator on 6 random polynomial triples (deg<=3) all zero: True
I1 n=4 Jacobiator on all coordinate triples zero (symbolic lam): True
I1 n=4 Jacobiator on 3 random polynomial triples (deg<=2) zero: True
I2 control (non-Poisson quadratic bivector) Jacobiator(x,y,z) = 2*x**2*y + 2*x*z**2 + 2*y**2*z (nonzero expected)
I3 A*K = [0, 0, 0]
I3 {f,g} == det[grad f, grad g, K] on 8 random pairs: True
I3 rank of A at (1,1,1) with (a,b,c)=(1,sqrt2,sqrt3): 2
I3 control (swapped kernel vector) difference nonzero: True
I4 minpoly(sqrt2+sqrt3) = t**4 - 10*t**2 + 1
I4 minpoly(2^(1/6)) = t**6 - 2 ; factor_list(t^6-2) = (1, [(t**6 - 2, 1)])
I4 degree of sqrt2+sqrt3 over Q: 4
I4 pslq(1,sqrt2,sqrt3) maxcoeff 10^6: None
I4 control pslq(1,sqrt2,1+sqrt2): [1, 1, -1]
I5 Pf(lam) = 2**(5/6) = 2**(5/6)
I5 det A = 2*2**(2/3)*x1**2*x2**2*x3**2*x4**2
I5 det(lam) = 2*2**(2/3) (should be Pf^2 = 2^(5/3))
I5 Jacobi on coordinate triples: True
I6 phi = x + 2*y - z : Jacobiator(x,y,z) = 0
I6 phi = x**2 + y*z : Jacobiator(x,y,z) = 0
I6 control n=4, phi=x1+x2: Jacobiator(x1,x3,x4) nonzero (rescaling trick is 3-dim only): True
