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add verification of computed valuation

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narodnik
2022-07-30 08:54:42 +02:00
parent 6607849889
commit a98f710217
1 changed files with 44 additions and 32 deletions
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76
script/research/ec/valuate.sage
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@@ -5,7 +5,7 @@ from tabulate import tabulate
# ord_P(y - 2x) = 2
# from Washington example 11.4 page 345
K.<x, y> = Integers(11)[]
K.<x, y> = GF(11)[]
Px, Py = K(2), K(4)
assert (3*Px^2 + 4) / (2*Py) == 2
@@ -24,8 +24,8 @@ def decomp(f, basis):
def comp(comps, basis):
return sum(a*b for a, b in zip(comps, basis))
f = y - 2*x
assert comp(decomp(f, basis), basis) == f
original_f = y - 2*x
assert comp(decomp(original_f, basis), basis) == original_f
# P = (a, b)
# y² = x³ + Ax + B
@@ -45,22 +45,22 @@ EC_B = 0
EC = y^2 - x^3 - A*x - B
# so we can replace (y - Py) with this
sub_poly_f = b0^2 + binomial(3,2)*Px*b0^1 + (3*Px^2 + EC_A)
sub_poly_g = (y + Py)
assert EC == b1*sub_poly_g - b0*sub_poly_f
Ef = b0^2 + binomial(3,2)*Px*b0^1 + (3*Px^2 + EC_A)
Eg = (y + Py)
assert EC == b1*Eg - b0*Ef
# f / g
# Technically we don't need g but we keep track of it anyway
def apply_reduction(comp_f, comp_g, basis):
#a1 = comp_f[1]
#comp_f[1] = 0
def apply_reduction(f, g, basis):
#a1 = f[1]
#f[1] = 0
b0, b1, _ = basis
# b1 == b0 * f / g
# so we can replace c b1 with (cf/g) b0
# a2 = 0
assert comp_f[2] == 0
assert f[2] == 0
# note that
# b1 = (f/g) b0
# so
@@ -71,49 +71,61 @@ def apply_reduction(comp_f, comp_g, basis):
# ----------- b0
# qg
comp_f[0] = comp_f[0]*sub_poly_g + comp_f[1]*sub_poly_f
comp_g[2] *= sub_poly_g
f[0] = f[0]*Eg + f[1]*Ef
f[1] = 0
g[0] *= Eg
k = 1
table = []
table.append(("", "f", "g", "k"))
def log(step_name, comp_f, comp_g, k):
table.append((step_name, str(comp_f), str(comp_g), k))
def log(step_name, f, g, k):
table.append((step_name, str(f), str(g), k))
comp_f = decomp(f, basis)
comp_g = [0, 0, 1]
log("start", comp_f, comp_g, k)
f = decomp(original_f, basis)
g = [1]
log("start", f, g, k)
# Reduce
apply_reduction(comp_f, comp_g, basis)
log("reduce", comp_f, comp_g, k)
apply_reduction(f, g, basis)
log("reduce", f, g, k)
f = comp_f[0]
f = f[0]
# Decompose
comp_f = decomp(f, basis)
comp_g = [0, 0, 1]
log("decomp", comp_f, comp_g, k)
f = decomp(f, basis)
log("decomp", f, g, k)
assert comp(comp_f, basis) == (x - 2)^2 - 5*(x - 2) - 2*(y - 4)
assert comp_f[2] == 0
assert comp(f, basis) == (x - 2)^2 - 5*(x - 2) - 2*(y - 4)
assert f[2] == 0
k += 1
# Reduce
apply_reduction(comp_f, comp_g, basis)
log("reduce", comp_f, comp_g, k)
apply_reduction(f, g, basis)
log("reduce", f, g, k)
f = comp_f[0]
f = f[0]
# Decompose
comp_f = decomp(f, basis)
comp_g = [0, 0, 1]
log("decomp", comp_f, comp_g, k)
f = decomp(f, basis)
log("decomp", f, g, k)
# Program terminates because remainder is nonzero
assert comp_f[2] != 0
assert f[2] != 0
print(f"basis = {basis}")
print(tabulate(table))
print(f"k = {k}")
# Test final value is correct
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
f0, f1, f2 = f
f = f0*b0 + f1*b1 + f2*b2
g = g[0]
fprime = b0^k * f/g
assert fprime == S(original_f)
# to convert fprime back again:
#f, g = fprime.numerator().lift(), fprime.denominator().lift()
assert g(Px, Py) != 0
assert f(Px, Py) != 0
assert b0(Px, Py) == 0