67 lines
2.8 KiB
Python
67 lines
2.8 KiB
Python
def decomposition_g0_g8(fct, prec = 50):
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'''Writes fct as a difference g0 - g8 + f, with g0 regular on the affine patch and g8 at the points in infinity
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and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
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C = fct.curve
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g = C.genus()
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coord = fct.coordinates(prec=prec)
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nontrivial_part = 0*C.x
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for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
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nontrivial_part += coord[i]*a
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fct -= nontrivial_part
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Fxy, Rxy, x, y = C.fct_field
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fct = Fxy(fct.function)
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num = fct.numerator()
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den = fct.denominator()
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integral_part, num = num.quo_rem(den)
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aux_den = superelliptic_function(C, Rxy(den))
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g0 = superelliptic_function(C, integral_part)
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g8 = superelliptic_function(C, 0)
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for monomial in num.monomials():
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aux = superelliptic_function(C, monomial)
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if aux.expansion_at_infty().valuation() >= aux_den.expansion_at_infty().valuation():
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g8 -= num.monomial_coefficient(monomial)*aux/aux_den
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else:
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g0 += num.monomial_coefficient(monomial)*aux/aux_den
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return (g0, g8, nontrivial_part)
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def decomposition_omega0_omega8(omega, prec=50):
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'''Writes omega as a difference omega0 - omega8, with omega0 regular on the affine patch and omega8 at the points in infinity.'''
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C = omega.curve
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omega.form = reduction(C, omega.form)
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F = C.base_ring
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delta = C.nb_of_pts_at_infty
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m = C.exponent
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if sum(omega.residue(place = i, prec = prec) for i in range(delta)) != 0:
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raise ValueError(str(omega) + " has non zero residue!")
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Fxy, Rxy, x, y = C.fct_field
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Rx.<x> = PolynomialRing(F)
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Fx = FractionField(Rx)
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fct = Fxy(omega.form)
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num = fct.numerator()
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den = fct.denominator()
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aux_den = superelliptic_function(C, Rxy(den))
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g0 = superelliptic_function(C, 0)
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g8 = superelliptic_function(C, 0)
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for j in range(0, m):
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component = Fx(omega.jth_component(j))
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q, r = component.numerator().quo_rem(component.denominator())
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g0 += (C.y)^(-j)*superelliptic_function(C, Rxy(q))
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if ((C.y)^(-j)*superelliptic_function(C, Fxy(r/component.denominator()))*C.dx).expansion_at_infty().valuation() < 0:
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raise ValueError("Something went wrong for "+str(omega))
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g8 -= (C.y)^(-j)*superelliptic_function(C, Fxy(r/component.denominator()))
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g0, g8 = g0*C.dx, g8*C.dx
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if g0.is_regular_on_U0():
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return (g0, g8)
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#Rx.<x> = PolynomialRing(F)
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#Rx.<x> = PolynomialRing(F)
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#aux_fct = (g0.form)*y
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else:
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raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))
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def decomposition_g0_g8_pth_power(fct):
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'''Decompose fct as g0 - g8 + A^p, if possible. Output: (g0, g8, A).'''
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coor = fct.coordinates()
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C = fct.curve
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return 0 |