49 lines
2.1 KiB
Python
49 lines
2.1 KiB
Python
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def decomposition_g0_pth_power(fct):
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'''Decompose fct as g0 + A^p, if possible. Output: (g0, A).'''
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omega = fct.diffn().regular_form()
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g0 = omega.int()
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A = (fct - g0).pth_root()
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return (g0, A)
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def decomposition_g0_p2th_power(fct):
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'''Decompose fct as g0 + A^(p^2), if possible. Output: (g0, A).'''
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g0, A = decomposition_g0_pth_power(fct)
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A0, A1 = decomposition_g0_pth_power(A)
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return (g0 + A0^p, A1)
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def decomposition_omega0_hpdh(omega):
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'''Decompose omega = (regular on U0) + h^(p-1) dh, so that Cartier(omega) = (regular on U0) + dh.
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Result: (regular on U0, h)'''
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omega1 = omega.cartier().cartier()
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omega1 = omega1.inv_cartier().inv_cartier()
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fct = (omega.cartier() - omega1.cartier()).int()
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return (omega1, fct)
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def decomposition_omega8_hpdh(omega, prec = 50):
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'''Decompose omega = (regular on U8) + h^(p-1) dh, so that Cartier(omega) = (regular on U8) + dh.
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Result: (regular on U8, h)'''
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C = omega.curve
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g = C.genus()
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Fxy, Rxy, x, y = C.fct_field
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F = C.base_ring
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p = C.characteristic
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Rt.<t> = LaurentSeriesRing(F)
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RT.<T> = PolynomialRing(F)
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FT = FractionField(RT)
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omega_analytic = FT(laurent_analytic_part(omega.expansion_at_infty(prec = prec))(t = T))
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print('omega_analytic', omega_analytic)
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Cv = C.uniformizer()
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v = Fxy(Cv.function)
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omega_analytic = Fxy(omega_analytic(T = v))
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print('expansions', superelliptic_function(C, omega_analytic).expansion_at_infty(prec = prec), '\n', Cv.diffn().expansion_at_infty(prec = prec),
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'\n', (superelliptic_function(C, omega_analytic)*Cv.diffn()).expansion_at_infty(prec = prec))
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omega_analytic = superelliptic_function(C, omega_analytic)*Cv.diffn()
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print('omega_analytic.expansion_at_infty()', omega_analytic.expansion_at_infty(prec = prec))
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print('omega_analytic', omega_analytic)
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omega8 = omega - omega_analytic
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print('omega8', omega8)
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dh = omega.cartier() - omega8.cartier()
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print('dh', dh)
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h = dh.int()
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print('omega8.expansion_at_infty()', omega8.expansion_at_infty(prec = prec))
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return (omega8, h)
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