PF ideals and their power series solutions

tests/test_period_computation.py

@@ -99,3 +99,75 @@ def test_simplify_factorises_theta_polynomial_per_z_monomial():
     # The underlying (expanded) operator is unchanged - only the display string differs.
     assert simplified.operator == op.operator
     assert simplified.operator_string == "-5*(5*theta0 + 4)*(5*theta0 + 3)*(5*theta0 + 2)*(5*theta0 + 1)*z0 + theta0^4"
+
+
+def test_find_power_series_solution_recovers_quintic_period():
+    # The quintic's Picard-Fuchs operator has indicial equation theta0^4 = 0 at z0 = 0
+    # (a quadruple root at 0), so its holomorphic (non-logarithmic) power series solution
+    # is found at indicial 0. Up to normalisation this is the classic quintic period
+    # 1 + 120*z0 + 113400*z0^2 + ...
+    op = PFOperator(
+        "theta0^4 - 5*z0*(5*theta0 + 1)*(5*theta0 + 2)*(5*theta0 + 3)*(5*theta0 + 4)",
+        no_variables=1,
+    )
+    ideal = PFIdeal([op])
+
+    solutions = ideal.find_power_series_solution(indicials=[0], order=3)
+
+    assert len(solutions) == 1
+    assert solutions[0].coefficients == {(0,): {(0,): 1, (1,): 120, (2,): 113400, (3,): 168168000}}
+
+
+def test_find_power_series_solution_handles_rational_indicial():
+    # theta0 - 1/2 kills z0^(1/2 + k) only when k = 0, since its eigenvalue is 1/2 + k;
+    # so at indicial 1/2 there is exactly one solution (a constant multiple of sqrt(z0)),
+    # while at indicial 0 no power series solution exists at all.
+    op = PFOperator("theta0 - 1/2", no_variables=1)
+    ideal = PFIdeal([op])
+
+    solutions = ideal.find_power_series_solution(indicials=[1/2], order=3)
+    assert len(solutions) == 1
+    assert solutions[0].coefficients == {(0,): {(0,): 1, (1,): 0, (2,): 0, (3,): 0}}
+    assert solutions[0].period_string == "sqrt(z0)"
+
+    assert ideal.find_power_series_solution(indicials=[0], order=3) == []
+
+
+def test_ideal_finds_holomorphic_solution_and_recovers_first_operator():
+    # Ideal generated by two operators (paper's z1, z2, theta1, theta2 <-> code's z0, z1,
+    # theta0, theta1):
+    #   M1 = theta2*(-2*theta1+2*theta2-1) + 2*(theta1-2*theta2-1)*(theta1-2*theta2)*z2
+    #   M2 = theta1^2*(2*(theta1-2*theta2)*z2-theta2) - 16*(2*theta1+1)*(4*theta1+1)*(4*theta1+3)*z1*z2
+    M1 = PFOperator(
+        "theta1*(-2*theta0 + 2*theta1 - 1) + 2*(theta0 - 2*theta1 - 1)*(theta0 - 2*theta1)*z1",
+        no_variables=2,
+    )
+    M2 = PFOperator(
+        "theta0^2*(2*(theta0 - 2*theta1)*z1 - theta1) - 16*(2*theta0 + 1)*(4*theta0 + 1)*(4*theta0 + 3)*z0*z1",
+        no_variables=2,
+    )
+    ideal = PFIdeal([M1, M2])
+
+    solutions = ideal.find_power_series_solution(indicials=[0, QQ(1) / 2], order=6)
+    assert len(solutions) == 1
+
+    period = solutions[0]
+    assert period.apply_operator(M1).coefficients == {}
+    assert period.apply_operator(M2).coefficients == {}
+    # Normalisation convention: the free parameter at the leading (0, 0) coefficient is 1.
+    assert period.coefficients[(0, 0)][(0, 0)] == 1
+    assert period.coefficients[(0, 0)][(1, 1)] == -32
+
+    # M1 lives entirely within z_degree <= 1, theta_degree <= 2 (a single bare factor of
+    # z2, quadratic in theta), so this is the natural Ansatz level to look for it at.
+    recovered = period.find_annihilating_operators(z_degree=1, theta_degree=2)
+    assert len(recovered) == 1
+
+    # recovered[0] should be a scalar multiple of M1 - compare via the coefficient of the
+    # bare theta2 (theta1 in code) monomial, which is nonzero in M1.
+    ratio = (
+        recovered[0].operator.monomial_coefficient(M1.theta_gens[1])
+        / M1.operator.monomial_coefficient(M1.theta_gens[1])
+    )
+    assert ratio != 0
+    assert recovered[0].operator == ratio * M1.operator