DeRhamComputation/superelliptic.ipynb

class superelliptic:
    def __init__(self, f, m, p):
        R.<x> = PolynomialRing(GF(p))
        self.polynomial = R(f)
        self.exponent = m
        self.characteristic = p
        
    def __repr__(self):
        f = self.polynomial
        m = self.exponent
        p = self.characteristic
        return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'
    
    def genus(self):
        r = self.polynomial.degree()
        m = self.exponent
        delta = GCD(r, m)
        return 1/2*((r-1)*(m-1) - delta + 1)
    
    def basis_holomorphic_differentials(self, j = 'all'):
        f = self.polynomial
        m = self.exponent
        p = self.characteristic
        r = f.degree()
        delta = GCD(r, m)
        
        basis = {}
        if j == 'all':
            k = 0
            for j in range(1, m):
                for i in range(1, r):
                    if (r*j - m*i >= delta):
                        basis[k] = superelliptic_form(C, x^(i-1)/y^j)
                        k = k+1
            return basis
        else:
            k = 0
            for i in range(1, r):
                if (r*j - m*i >= delta):
                    basis[k] = superelliptic_form(C, x^(i-1)/y^j)
                    k = k+1
            return basis
     
    def basis_de_rham(self, j = 'all'):
        f = self.polynomial
        m = self.exponent
        p = self.characteristic
        r = f.degree()
        delta = GCD(r, m)
        #?????
        basis = {}
        if j == 'all':
            for j in range(1, m):
                holo = baza_holo(m, f, j, p)
    for k in range(0, len(holo)):
        baza[k] = holo[k]
    
def reduction(C, g):
    p = C.characteristic
    R.<x, y> = PolynomialRing(GF(p), 2)
    RR = FractionField(R)
    f = C.polynomial
    r = f.degree()
    m = C.exponent
    g = RR(g)
    g1 = g.numerator()
    g2 = g.denominator()
    
    R1.<x> = PolynomialRing(GF(p))
    R2 = FractionField(R1)
    R3.<y> = PolynomialRing(R2)    
    (A, B, C) = xgcd(R3(g2), R3(y^m - f))
    g = R3(g1*B/A)
    
    while(g.degree(R(y)) >= m):
        d = g.degree(R(y))
        G = coff(g, d)
        i = floor(d/m)
        g = g - G*y^d + f^i * y^(d%m) *G
    
    return(R3(g))

def reduction_form(C, g):
    p = C.characteristic
    R.<x, y> = PolynomialRing(GF(p), 2)
    RR = FractionField(R)
    f = C.polynomial
    r = f.degree()
    m = C.exponent
    g = reduction(C, g)

    g1 = RR(0)
    R1.<x> = PolynomialRing(GF(p))
    R2 = FractionField(R1)
    R3.<y> = PolynomialRing(R2)
    
    g = R3(g)
    for j in range(0, m):
        if j==0:
            G = coff(g, 0)
            g1 += G
        else:
            G = coff(g, j)
            g1 += RR(y^(j-m)*f*G)
    return(g1)
        
class superelliptic_function:
    def __init__(self, C, g):
        R.<x, y> = PolynomialRing(GF(p), 2)
        RR = FractionField(R)
        f = C.polynomial
        r = f.degree()
        m = C.exponent
        
        self.curve = C
        g = reduction(C, g)
        self.function = g
        
    def __repr__(self):
        return str(self.function)
    
    def jth_component(self, j):
        g = self.function
        R.<x, y> = PolynomialRing(GF(p), 2)
        g = R(g)
        return g.coefficient(y^j)
    
    def __add__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 + g2)
        return superelliptic_function(C, g)
    
    def __sub__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 - g2)
        return superelliptic_function(C, g)
    
    def __mul__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 * g2)
        return superelliptic_function(C, g)
    
    def __truediv__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 / g2)
        return superelliptic_function(C, g)
    
def diffn(self):
    C = self.curve
    f = C.polynomial
    m = C.exponent
    g = self.function
    A = g.derivative(x)
    B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
    return superelliptic_form(C, A+B)
        
class superelliptic_form:
    def __init__(self, C, g):
        R.<x, y> = PolynomialRing(GF(p), 2)
        RR = FractionField(R)
        g = RR(reduction_form(C, g))
        self.form = g
        self.curve = C      
        
    def __add__(self, other):
        C = self.curve
        g1 = self.form
        g2 = other.form
        g = reduction(C, g1 + g2)
        return superelliptic_form(C, g)
    
    def __sub__(self, other):
        C = self.curve
        g1 = self.form
        g2 = other.form
        g = reduction(C, g1 - g2)
        return superelliptic_form(C, g)
    
    def __repr__(self):
        g = self.form
        if len(str(g)) == 1:
            return str(g) + ' dx'
        return '('+str(g) + ') dx'
    
    def jth_component(self, j):
        g = self.form
        R1.<x> = PolynomialRing(GF(p))
        R2 = FractionField(R1)
        R3.<y> = PolynomialRing(R2)
        R4 = FractionField(R3)
        R5.<y_inv> = PolynomialRing(R2)
        g = R4(g)
        g = g(y = 1/y_inv)
        g = R5(g)
        return coff(g, j)
    
    def is_regular_on_U0(self):
        C = self.curve
        p = C.characteristic
        m = C.exponent
        R.<x> = PolynomialRing(GF(p))
        for j in range(1, m):
            if self.jth_component(j) not in R:
                return 0
        return 1
    
    def is_regular_on_Uinfty(self):
        C = self.curve
        p = C.characteristic
        m = C.exponent
        f = C.polynomial
        r = f.degree()
        delta = GCD(m, r)
        M = m/delta
        R = r/delta
        
        for j in range(1, m):
            A = self.jth_component(j)
            d = degree_of_rational_fctn(A)
            if(-d*M + j*R -(M+1)<0):
                return 0
        return 1
    
class superelliptic_cech:
    def __init__(self, omega, fct):
        self.omega0 = omega
        self.omega8 = omega - diffn(fct)
        self.f = fct
        
def degree_of_rational_fctn(f):
    R.<x> = PolynomialRing(GF(p))
    RR = FractionField(R)
    f = RR(f)
    f1 = f.numerator()
    f2 = f.denominator()
    d1 = f1.degree()
    d2 = f2.degree()
    return(d1 - d2)

def coff(f, d):
    lista = f.coefficients(sparse = false)
    if len(lista) <= d:
        return 0
    return lista[d]

def cut(f, i):
    coeff = f.coefficients(sparse = false)
    return sum(x^(j-i-1) * coeff[j] for j in range(i+1, f.degree() + 1))

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-87-fb238ae380ad> in <module>()
----> 1 class superelliptic:
      2     def __init__(self, f, m, p):
      3         R = PolynomialRing(GF(p), names=('x',)); (x,) = R._first_ngens(1)
      4         self.polynomial = R(f)
      5         self.exponent = m

<ipython-input-87-fb238ae380ad> in superelliptic()
     53             for j in range(Integer(1), m):
     54                 holo = baza_holo(m, f, j, p)
---> 55     for k in range(Integer(0), len(holo)):
     56         baza[k] = holo[k]
     57 

NameError: name 'holo' is not defined

C = superelliptic(x^3 + x + 2, 2, 5)
omega = superelliptic_form(C, (x^2+x)/y + 3*x^3)
omega

((-2*x^3*y + x^2 + x)/y) dx

licz = 0
m = 2
p = 5
R1.<x> = PolynomialRing(GF(p))
f = R1(x^3 + x + 4)
r = f.degree()
C = superelliptic(f, m, p)
for i in range(0, r):
    for j in range(1, m):
        omega = superelliptic_form(C, x^i/y^j)
        if (omega.is_regular_on_U0() and omega.is_regular_on_Uinfty()):
            print(omega)
            licz += 1
print(licz, C.genus())
print(C.basis_holomorphic_differentials())

(1/y) dx
1 1
{0: (1/y) dx}

p = 5
R.<x, y> = PolynomialRing(GF(p), 2)
g = x^6*y^2 + y^2

omega = diffn(superelliptic_function(C, y^2))

omega.jth_component(0)

3*x^2 + 1

R.<x, y> = PolynomialRing(GF(p), 2)
g1 = x^3*y^7 + x^2*y^9
g2 = x^2*y + y^6
R1.<x> = PolynomialRing(GF(p))
R2 = FractionField(R1)
R3.<y> = PolynomialRing(R2)

xgcd(R3(g1), R3(g2))[1]*R3(g1) + xgcd(R3(g1), R3(g2))[2]*R3(g2)

y

H = HyperellipticCurve(x^5 - x + 1)

H

Hyperelliptic Curve over Finite Field of size 5 defined by y^2 = x^5 + 4*x + 1

f = x^3 + x + 2

f.derivative(x)

-2*x^2 + 1

p = 5
R1.<x> = PolynomialRing(GF(p))
R2 = FractionField(R1)
R3.<y> = PolynomialRing(R2)
g = y^2/x + y/(x+1)    
g = 1/y+x/y^2

R3.<z> = PolynomialRing(R2)
g(y = 1/z)

x*z^2 + z

f

x^3 + x + 4

f.coefficient()

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-62-e054c182ec1a> in <module>()
----> 1 f.coefficient()

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4614)()
    485             AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah'
    486         """
--> 487         return self.getattr_from_category(name)
    488 
    489     cdef getattr_from_category(self, name):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4723)()
    498         else:
    499             cls = P._abstract_element_class
--> 500         return getattr_from_other_class(self, cls, name)
    501 
    502     def __dir__(self):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/cpython/getattr.pyx in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2614)()
    392         dummy_error_message.cls = type(self)
    393         dummy_error_message.name = name
--> 394         raise AttributeError(dummy_error_message)
    395     attribute = <object>attr
    396     # Check for a descriptor (__get__ in Python)

AttributeError: 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint' object has no attribute 'coefficient'