cohomology of str sheaf dodany; zaczynamy zmieniac wspolrzedne w superelliptic holo

sage/.run.term-0.term

@@ -26748,4 +26748,133 @@ Input In [81], in <cell line: 1>()
 ]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add draft/supsage/drafty/superelliptic_cohomology_class.sage 
 ]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ gid add -u
 bash: gid: command not found
-]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ gid add -u
+]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ gid add -u[1@t
+]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git commit -m ""C"o"h"o"m"o"l"o"g"y" "o"f" "s"t"r"u"c"t"u"r"e" "s"h"e"a"f" "o"f" "s"u"p"e"r"e"l"l"i"p"t"i"c"
+[master be66e7b] Cohomology of structure sheaf of superelliptic
+ 11 files changed, 7056 insertions(+), 82 deletions(-)
+ delete mode 100644 sage/as_covers/as_cover/uniformizer.sage
+ rewrite sage/drafty/draft3.sage (90%)
+ create mode 100644 sage/drafty/superelliptic_cohomology_class.sage
+]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git push
+Username for 'https://git.wmi.amu.edu.pl': jgarnek
+Password for 'https://jgarnek@git.wmi.amu.edu.pl': 
+Enumerating objects: 37, done.
+Counting objects:   2% (1/37)
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Counting objects: 100% (37/37), done.
+Delta compression using up to 4 threads
+Compressing objects:   4% (1/23)
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Compressing objects: 100% (23/23), done.
+Writing objects:   4% (1/23)
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Writing objects: 100% (23/23), 73.16 KiB | 337.00 KiB/s, done.
+Total 23 (delta 15), reused 0 (delta 0)
+remote: . Processing 1 references
+remote: Processed 1 references in total
+To https://git.wmi.amu.edu.pl/jgarnek/DeRhamComputation.git
+   d77adde..be66e7b  master -> master
+]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ ]0;~/Research/2021 De Rham/DeRhamComputation/sage~/Research/2021 De Rham/DeRhamComputation/sage$ 
]0;~/Research/2021 De Rham/DeRhamComputation/sage~/Research/2021 De Rham/DeRhamComputation/sage$ sage
+┌────────────────────────────────────────────────────────────────────┐
+│ SageMath version 9.7, Release Date: 2022-09-19                     │
+│ Using Python 3.10.5. Type "help()" for help.                       │
+└────────────────────────────────────────────────────────────────────┘
+]0;IPython: DeRhamComputation/sage[?2004h[?1l[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lload('init.sage')[?7h[?12l[?25h[?25l[?7lo[?7h[?12l[?25h[?25l[?7lad('init.sage')[?7h[?12l[?25h[?25l[?7lsage: load('init.sage')
+[?7h[?12l[?25h[?2004l[5]
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC.is_smooth()[?7h[?12l[?25h[?25l[?7lsage: C
+[?7h[?12l[?25h[?2004l[?7hSuperelliptic curve with the equation y^4 = x^5 + x over Finite Field of size 3
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC[?7h[?12l[?25h[?25l[?7l.is_smooth()[?7h[?12l[?25h[?25l[?7lf[?7h[?12l[?25h[?25l[?7lr[?7h[?12l[?25h[?25l[?7lo[?7h[?12l[?25h[?25l[?7lbenius_matrix[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l([?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7lsage: C.frobenius_matrix()
+[?7h[?12l[?25h[?2004l---------------------------------------------------------------------------
+TypeError                                 Traceback (most recent call last)
+Input In [3], in <cell line: 1>()
+----> 1 C.frobenius_matrix()
+
+File <string>:163, in frobenius_matrix(self, prec)
+
+TypeError: unsupported operand type(s) for ** or pow(): 'superelliptic_cohomology' and 'method'
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC.frobenius_matrix()[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lload('init.sage')[?7h[?12l[?25h[?25l[?7lsage: load('init.sage')
+[?7h[?12l[?25h[?2004l[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC.frobenius_matrix()[?7h[?12l[?25h[?25l[?7l.[?7h[?12l[?25h[?25l[?7lf[?7h[?12l[?25h[?25l[?7lr[?7h[?12l[?25h[?25l[?7lo[?7h[?12l[?25h[?25l[?7lbenius_matrix()[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7lp)[?7h[?12l[?25h[?25l[?7lr)[?7h[?12l[?25h[?25l[?7le)[?7h[?12l[?25h[?25l[?7lc)[?7h[?12l[?25h[?25l[?7l-)[?7h[?12l[?25h[?25l[?7l5)[?7h[?12l[?25h[?25l[?7l0)[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7l-)[?7h[?12l[?25h[?25l[?7l5)[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7l=)[?7h[?12l[?25h[?25l[?7l5)[?7h[?12l[?25h[?25l[?7l0)[?7h[?12l[?25h[?25l[?7lsage: C.frobenius_matrix(prec=50)
+[?7h[?12l[?25h[?2004l---------------------------------------------------------------------------
+TypeError                                 Traceback (most recent call last)
+Input In [5], in <cell line: 1>()
+----> 1 C.frobenius_matrix(prec=Integer(50))
+
+File <string>:163, in frobenius_matrix(self, prec)
+
+TypeError: unsupported operand type(s) for ** or pow(): 'superelliptic_cohomology' and 'method'
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lload('init.sage')[?7h[?12l[?25h[?25l[?7load('init.sage')[?7h[?12l[?25h[?25l[?7lsage: load('init.sage')
+[?7h[?12l[?25h[?2004l[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lload('init.sage')[?7h[?12l[?25h[?25l[?7lC.frobenius_matrix(prec=50)[?7h[?12l[?25h[?25l[?7lsage: C.frobenius_matrix(prec=50)
+[?7h[?12l[?25h[?2004l---------------------------------------------------------------------------
+TypeError                                 Traceback (most recent call last)
+File /ext/sage/9.7/src/sage/rings/polynomial/polynomial_element.pyx:2441, in sage.rings.polynomial.polynomial_element.Polynomial.__pow__()
+   2440 try:
+-> 2441     right = Integer(right)
+   2442 except TypeError:
+
+File /ext/sage/9.7/src/sage/rings/integer.pyx:717, in sage.rings.integer.Integer.__init__()
+    716 
+--> 717                 raise TypeError("unable to coerce %s to an integer" % type(x))
+    718 
+
+TypeError: unable to coerce <class 'method'> to an integer
+
+During handling of the above exception, another exception occurred:
+
+TypeError                                 Traceback (most recent call last)
+Input In [7], in <cell line: 1>()
+----> 1 C.frobenius_matrix(prec=Integer(50))
+
+File <string>:163, in frobenius_matrix(self, prec)
+
+File <string>:71, in __pow__(self, exp)
+
+File /ext/sage/9.7/src/sage/rings/polynomial/polynomial_element.pyx:2443, in sage.rings.polynomial.polynomial_element.Polynomial.__pow__()
+   2441         right = Integer(right)
+   2442     except TypeError:
+-> 2443         raise TypeError("non-integral exponents not supported")
+   2444 
+   2445 d = self.degree()
+
+TypeError: non-integral exponents not supported
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l((C.x)*C.x*C.dx).cartier()[?7h[?12l[?25h[?25l[?7lC.one/[?7h[?12l[?25h[?25l[?7l.[?7h[?12l[?25h[?25l[?7lx*C.dx).cartier()[?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7l()^[?7h[?12l[?25h[?25l[?7lp[?7h[?12l[?25h[?25l[?7lsage: (C.x)^p
+[?7h[?12l[?25h[?2004l[?7hx^7
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC.frobenius_matrix(prec=50)[?7h[?12l[?25h[?25l[?7lsage: C
+[?7h[?12l[?25h[?2004l[?7hSuperelliptic curve with the equation y^4 = x^5 + x over Finite Field of size 3
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lprint(len(lista))[?7h[?12l[?25h[?25l[?7lsage: p
+[?7h[?12l[?25h[?2004l[?7h7
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lload('init.sage')[?7h[?12l[?25h[?25l[?7lo[?7h[?12l[?25h[?25l[?7lad('init.sage')[?7h[?12l[?25h[?25l[?7lsage: load('init.sage')
+[?7h[?12l[?25h[?2004l[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC[?7h[?12l[?25h[?25l[?7l.frobenius_matrix(prec=50)[?7h[?12l[?25h[?25l[?7lfrobenius_matrix(prec=50)[?7h[?12l[?25h[?25l[?7lsage: C.frobenius_matrix(prec=50)
+[?7h[?12l[?25h[?2004l[?7h[0 0 0 0 0 1]
+[0 0 1 0 0 0]
+[0 1 0 0 0 0]
+[1 0 0 0 0 0]
+[0 0 0 0 0 0]
+[0 0 0 0 0 0]
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC.frobenius_matrix(prec=50)[?7h[?12l[?25h[?25l[?7l.[?7h[?12l[?25h[?25l[?7ldx.cartier()[?7h[?12l[?25h[?25l[?7le_rham_basis()[?7h[?12l[?25h[?25l[?7l_rham_basis()[?7h[?12l[?25h[?25l[?7l([?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lb[?7h[?12l[?25h[?25l[?7la[?7h[?12l[?25h[?25l[?7ls[?7h[?12l[?25h[?25l[?7li[?7h[?12l[?25h[?25l[?7lsage: C.basis_
+ C.basis_de_rham_degrees                 
+ C.basis_holomorphic_differentials_degree
+ C.basis_of_cohomology                   
+
+ [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lde_rham_degrees
+ C.basis_de_rham_degrees                 
+
+
+ <unknown> [?7h[?12l[?25h[?25l[?7lholomorphic_differentials_degree
+ C.basis_de_rham_degrees                 
+ C.basis_holomorphic_differentials_degree[?7h[?12l[?25h[?25l[?7lof_cohomology
+
+ C.basis_holomorphic_differentials_degree
+ C.basis_of_cohomology                   [?7h[?12l[?25h[?25l[?7l
+
+
+
+[?7h[?12l[?25h[?25l[?7l([?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7lsage: C.basis_of_cohomology()
+[?7h[?12l[?25h[?2004l[?7h[1/x*y, 1/x*y^2, 1/x^2*y^2, 1/x*y^3, 1/x^2*y^3, 1/x^3*y^3]
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7lsage: 
+
+
+ [?7h[?12l[?25h[?25l[?7lC.basis_of_cohomology()[?7h[?12l[?25h[?25l[?7l()[[?7h[?12l[?25h[?25l[?7l0[?7h[?12l[?25h[?25l[?7l[][?7h[?12l[?25h[?25l[?7l[].[?7h[?12l[?25h[?25l[?7l[][?7h[?12l[?25h[?25l[?7l[]^[?7h[?12l[?25h[?25l[?7l[[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lp[?7h[?12l[?25h[?25l[?7l)[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[][?7h[?12l[?25h[?25l[?7l[][?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l(C.basis_of_cohomology()[0]^p)[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7l().[?7h[?12l[?25h[?25l[?7lc[?7h[?12l[?25h[?25l[?7lo[?7h[?12l[?25h[?25l[?7lo[?7h[?12l[?25h[?25l[?7lr[?7h[?12l[?25h[?25l[?7ld[?7h[?12l[?25h[?25l[?7li[?7h[?12l[?25h[?25l[?7ln[?7h[?12l[?25h[?25l[?7la[?7h[?12l[?25h[?25l[?7lt[?7h[?12l[?25h[?25l[?7le[?7h[?12l[?25h[?25l[?7ls[?7h[?12l[?25h[?25l[?7l([?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7lsage: (C.basis_of_cohomology()[0]^p).coordinates()
+[?7h[?12l[?25h[?2004l[?7h[0, 0, 0, 0, 1, 0]
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7lsage: 
+ [?7h[?12l[?25h[?25l[?7lp[?7h[?12l[?25h[?25l[?7lsage: p
+[?7h[?12l[?25h[?2004l[?7h7
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lC.basis_of_cohomology()[?7h[?12l[?25h[?25l[?7lsage: C
+[?7h[?12l[?25h[?2004l[?7hSuperelliptic curve with the equation y^4 = x^5 + x over Finite Field of size 3
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lp[?7h[?12l[?25h[?25l[?7l = 5[?7h[?12l[?25h[?25l[?7l=[?7h[?12l[?25h[?25l[?7l [?7h[?12l[?25h[?25l[?7l3[?7h[?12l[?25h[?25l[?7lsage: p = 3
+[?7h[?12l[?25h[?2004l[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lp = 3[?7h[?12l[?25h[?25l[?7lC[?7h[?12l[?25h[?25l[?7lp[?7h[?12l[?25h[?25l[?7l(C.basis_of_cohomology()[0]^p).coordinates()[?7h[?12l[?25h[?25l[?7lsage: (C.basis_of_cohomology()[0]^p).coordinates()
+[?7h[?12l[?25h[?2004l[?7h[0, 0, 0, 0, 0, 1]
+[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lq = 5[?7h[?12l[?25h

sage/init.sage

@@ -18,7 +18,7 @@ load('auxilliaries/linear_combination_polynomials.sage')
 ##############
 ##############
 load('drafty/lift_to_de_rham.sage')
-load('drafty/superelliptic_cohomology_class.sage')
+#load('drafty/superelliptic_cohomology_class.sage')
 load('drafty/draft5.sage')
 load('drafty/pole_numbers.sage')
 #load('drafty/draft4.sage')

sage/superelliptic/superelliptic_class.sage

@@ -12,6 +12,7 @@ class superelliptic:
         self.exponent = m
         self.base_ring = F
         self.characteristic = F.characteristic()
+        self.fct_field = RxyzQ, Rxyz, x, y, z
         r = Rx(f).degree()
         delta = GCD(r, m)
         self.nb_of_pts_at_infty = delta
@@ -26,6 +27,19 @@ class superelliptic:
         F = self.base_ring
         return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over ' + str(F)
 
+    def coordinates2(self, basis = 0):
+        """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
+        C = self.curve
+        if basis == 0:
+            basis = C.holomorphic_differentials_basis()
+        RxyzQ, Rxyz, x, y, z = C.fct_field
+        # We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
+        # and sometimes basis elements have denominators. Thus we multiply by them.
+        denom = LCM([denominator(omega.form) for omega in basis])
+        basis = [denom*omega for omega in basis]
+        self_with_no_denominator = denom*self
+        return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis]) 
+    
     #Auxilliary algorithm that returns the basis of holomorphic differentials
     #of the curve and (as a second argument) the list of pairs (i, j)
     #such that x^i dx/y^j is holomorphic.
@@ -128,7 +142,7 @@ class superelliptic:
             M[i, :] = v
         return M
     
-    def frobenius_matrix(self):
+    def dr_frobenius_matrix(self):
         basis = self.de_rham_basis()
         g = self.genus()
         p = self.characteristic
@@ -153,6 +167,16 @@ class superelliptic:
             M[i, :] = v
         return M     
 
+    def frobenius_matrix(self, prec=20):
+        g = self.genus()
+        F = self.base_ring
+        p = F.characteristic()
+        M = matrix(F, g, g)
+        for i, f in enumerate(self.basis_of_cohomology()):
+            M[i, :] = vector((f^p).coordinates(prec=prec))
+        M = M.transpose()
+        return M
+    
 #    def p_rank(self):
 #        return self.cartier_matrix().rank()
     
@@ -166,6 +190,23 @@ class superelliptic:
         p = self.characteristic
         return flag(Fr, V, p, test)
     
+    def basis_of_cohomology(self):
+        '''Basis of cohomology of structure sheaf H1(X, OX).'''
+        m = self.exponent
+        f = self.polynomial
+        r = f.degree()
+        F = self.base_ring
+        delta = self.nb_of_pts_at_infty
+        Rx.<x> = PolynomialRing(F)
+        Rxy.<x, y> = PolynomialRing(F, 2)
+        Fxy = FractionField(Rxy)
+        basis = []
+        for j in range(1, m):
+                for i in range(1, r):
+                    if (r*j - m*i >= delta):
+                        basis += [superelliptic_function(self, Fxy(m*y^(j)/x^i))]
+        return basis
+
 #Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
 #it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
 #you obtain \sum_{i = 0}^{m-1} y^i g_i(x).

sage/superelliptic/superelliptic_form_class.sage

@@ -54,6 +54,14 @@ class superelliptic_form:
             result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))
         return result   
 
+    def serre_duality_pairing(self, fct, prec=20):
+        result = 0
+        C = self.curve
+        delta = C.nb_of_pts_at_infty
+        for i in range(delta):
+            result += (fct*self).expansion_at_infty(place=i, prec=prec)[-1]
+        return -result
+    
     def coordinates(self):
         C = self.curve
         F = C.base_ring

sage/superelliptic/superelliptic_function_class.sage

@@ -83,6 +83,18 @@ class superelliptic_function:
         B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
         return superelliptic_form(C, A+B)
 
+    def coordinates(self, basis = 0, basis_holo = 0, prec=20):
+        '''Find coordinates in H1(X, OX) in given basis basis with dual basis basis_holo.'''
+        C = self.curve
+        if basis == 0:
+            basis = basis_of_cohomology(C)
+        if basis_holo == 0:
+            basis_holo = C.holomorphic_differentials_basis()
+        g = C.genus()
+        coordinates = g*[0]
+        for i, omega in enumerate(basis_holo):
+            coordinates[i] = omega.serre_duality_pairing(self, prec=prec)
+        return coordinates
     
     def expansion_at_infty(self, place = 0, prec=10):
         C = self.curve