21 KiB
21 KiB
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)
R2.<x, y> = PolynomialRing(GF(p), 2)
RR = FractionField(R2)
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(self, RR(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(self, RR(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)
R.<x> = PolynomialRing(GF(p))
R1.<x, y> = PolynomialRing(GF(p), 2)
RR = FractionField(R1)
basis = {}
if j == 'all':
for j in range(1, m):
holo = C.basis_holomorphic_differentials(j)
for k in range(0, len(holo)):
basis[k] = superelliptic_cech(self, holo[k], superelliptic_function(self, R(0)))
k = len(basis)
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = R(m-j)*R(x)*R(f.derivative()) - R(m)*R(i)*f
psi = R(cut(s, i))
basis[k] = superelliptic_cech(self, superelliptic_form(self, psi/y^j), superelliptic_function(self, m*y^j/x^i))
k = k+1
return basis
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):
p = C.characteristic
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
p = C.characteristic
g = self.function
R.<x, y> = PolynomialRing(GF(p), 2)
RR = FractionField(R)
g = RR(g)
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):
p = C.characteristic
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, C, omega, fct):
self.omega0 = omega
self.omega8 = omega - diffn(fct)
self.f = fct
self.curve = C
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):
R = f.parent()
coeff = f.coefficients(sparse = false)
return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))C = superelliptic(x^3 + x + 2, 2, 5)
C.basis_de_rham()
#C.basis_holomorphic_differentials(){0: <__main__.superelliptic_cech object at 0x6fdcefeaf60>,
1: <__main__.superelliptic_cech object at 0x6fdcefea8d0>}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^2omega = 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)HHyperelliptic Curve over Finite Field of size 5 defined by y^2 = x^5 + 4*x + 1
f = x^3 + x + 2f.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^2R3.<z> = PolynomialRing(R2)
g(y = 1/z)x*z^2 + z
fx^3 + x + 4
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x^3+x+1x^3 + x + 1
parent(x)Symbolic Ring
R.<x> = PolynomialRing(GF(5))R = (x^3+x).parent()R.<x, y> = PolynomialRing(GF(5))
RR = FractionField(R)
A = RR(1/(x*y))A.derivative(x)(-1)/(x^2*y)