Better tangent vector for orbit-closure --deform

[saraedum]

Currently, we deform with the first tangent vector we find, scaled so that it is shorter than (1, 0).

There are several problems with this approach:

• A tangent vector of the form (x, 0) often leads to lots of flips when calling FlatTriangulation::operator+. It is probably better to use a vector that is not axis-aligned. (Or explicitly search for a vector that does not need any flips.)
• The tangent vector often has huge coefficients. (Both in absolute size and in bitsize.) We should try to find a smaller vector without just dividing by some huge constant. More specifically, we should find a tangent vector that when added to the surface, leads to small coefficients.

Without these changes, we cannot seem to go systematically beyond triangles (a, b, c) with a + b + c > 60.

[saraedum]

The easiest approach might be:

• Write the tangent vectors as tuples in ZZ, i.e., as the integral coefficients of their number field coefficients/exact real coefficients.
• Use LLL to find a small vector v in this lattice.
• Do not deform if this vector is already in the span of the initial tangent space generated by the x and y coordinates of the triangulation.
• Deform by (v/2, v/4) or maybe by (v-x, v/2).

As discussed, it is unclear whether LLL can find anything useful here since the dimension of the tangent space is very small compared to number of half edges × number field degree. However, doing LLL over number fields seems to be a very hard problem, see e.g., https://link.springer.com/chapter/10.1007/978-3-030-34621-8_3.

[videlec]

• It would be useful to have a .find_simple_surface() operation on GL2ROrbitClosure that tries to find the "simplest" surface around. Namely one whose coefficients are as simple as possible. Having such operation might be useful for further deformation as this simplest surface might be a better starting point for deformations.
• Related to the above, a first step would be to implement the "field reduction". By definition there exists a surface in the orbit closure whose coefficients belongs to GL2ROrbitClosure.field_of_definition(). Though the latter method uses the echelon form which is dramatically expensive in general. I am not sure how it can be done without it. Assuming that we knew how to implement efficiently GL2ROrbitClosure.field_of_definition() it is not even clear to me how to find a surface with coefficients in that field.

[saraedum]

As a minimalistic first step, we are now deforming with the smallest vector in the tangent space (that does not come from the x,y coordinates) and only deform if the blowup is less than 1e20. (That's probably not a good bound.)