Implementing Frobenius basis for PFIdeals

README.md

@@ -8,7 +8,7 @@ It allows for computation of discriminant loci and topological data of manifolds
 Supported initialisations are represented by the following examples.
 
 ```python
-elliptic_curve_D = ToricPolytopeProjectiveSpace([1, 2, 3], model_name="elliptic_curve_D")
+elliptic_curve_D = ToricPolytopeProjectiveSpace([3, 2, 1], model_name="elliptic_curve_D")
 
 CY3_quintic = ToricPolytopeProjectiveSpace([1, 1, 1, 1, 1], model_name="quintic")
 CY3_bicubic = ToricPolytopeCICY([[3, 3]], model_name="bi-cubic")
@@ -149,4 +149,43 @@ recovered[0].simplify().operator_string
 ```
 
 `recovered[0]` is, up to scale, `M1` — recovered purely from `M1`, `M2`'s shared power series
-solution. 
+solution. 
+
+### Frobenius bases
+
+A `PFIdeal` also computes its complete Frobenius basis — all solutions, logarithmic ones included —
+at the origin of its current coordinates; for another point, move it to the origin with `change_coordinates`
+first. The exponents are the `indicials`, the roots of the system's indicial ideal, which is completed by linear
+algebra where the operators' z-free parts alone don't determine it. The optional `hodge_numbers` — the
+numbers of solutions of top log-degree 0, 1, ..., n at a MUM point, e.g. `[1, 2, 2, 1]` for a two-parameter
+threefold — certify that completion through their sum, the rank, and bound the powers of logarithms. For
+P_{2,2,2,1,1}[8] at its MUM point and at the point (z0, z1) = (0, 1/4) on its strong coupling divisor:
+
+```python
+D1 = PFOperator(
+    "theta0^2*(theta0 - 2*theta1) - 4*z0*(4*theta0 + 3)*(4*theta0 + 2)*(4*theta0 + 1)",
+    no_variables=2,
+)
+D2 = PFOperator("theta1^2 - z1*(2*theta1 - theta0 + 1)*(2*theta1 - theta0)", no_variables=2)
+ideal = PFIdeal([D1, D2], hodge_numbers=[1, 2, 2, 1])
+strong_coupling = ideal.change_coordinates("z0", "z1 - 1/4")
+
+ideal.indicials()
+strong_coupling.indicials()
+basis = strong_coupling.frobenius_basis(order=2)
+basis[5].period_string
+```
+
+```term
+[((0, 0), 6)]
+[((0, 0), 4), ((0, 1/2), 2)]
+INFO:__main__:Completed the indicial ideal at z = 0 at multiplier degree 2 with 1 extra operator(s): colength 6.
+INFO:__main__:Exponent class of (0, 0): 4 solution(s) with leading terms ['1', 'log(z0)', 'log(z0)^2', 'log(z0)^3'].
+INFO:__main__:Exponent class of (0, 1/2): 2 solution(s) with leading terms ['sqrt(z1)', 'sqrt(z1)*(log(z0) + log(z1))'].
+INFO:__main__:Found a Frobenius basis of 6 solution(s) at order 2.
+'-1/225*(7200*z0*z1*log(z0) - 720*z1^2*log(z0) + 7200*z0*z1*log(z1) - 720*z1^2*log(z1) + 33600*z0*z1 - 732*z1^2 + 300*z1*log(z0) + 300*z1*log(z1) + 250*z1 - 225*log(z0) - 225*log(z1))*sqrt(z1)'
+```
+
+Within each exponent class the basis is canonical: every solution has its own leading term with coefficient 1,
+at which the class' other solutions vanish. Coefficients are those of plain powers of log(z_i), without factors
+of 2πi.