Singular

D.15.36.5 SaitoBase

Usage:

SaitoBase(VecField V)

Return:

new VecField W in the base from [1] thm. 3.1, where semisimple and nilpotent components are easily read off.

Note:

the algorithm requires inversions of algebra morphisms. These will be exact to the precision of V. Warning: The algorithm assumes standard coordinates. If V is not given in standard coordinates, it will be converted to standard coordinates, but not converted back at the end, so the resulting transformation is from standard coordinates to the Saito coordinates. If you want the entire transformation from your original coordinates to the Saito ones, add the inverse of your original coordinate transformation manually.
Note that weight vectors only take Int64,
making the algorithm fail for very large entries in V.vec.

LIB "VecField.lib";
ring r = 0, (x, y, z),ds;
vector v = [-1,-1,-1];
VecField V = v;
V.precision = 4;
map phi = r, x-2y2+z3,2y+y3+z,z;
VecField W = changeCoordinates(V,phi);
VecField WS = SaitoBase(W);