def test__diff__1(self):
ring = PolyRing('x,y,v,w', True)
ring.ext_num_field('t^2 + 1')
x, y, v, w, a0 = ring.coerce('x,y,v,w,a0')
assert ring.diff(w - a0, w, 1) == 1
def test__diff__2(self):
ring = PolyRing('x,y,v,w', True)
ring.ext_num_field('t^2 + 1')
x, y, v, w, a0 = ring.coerce('x,y,v,w,a0')
assert ring.diff(x**3 - a0, x, 3) == 6
def test__quo( self ):
ring = PolyRing( 'x,y,z', True )
ring.ext_num_field( 't^2 + 1' )
ring.ext_num_field( 't^3 + a0' )
ls = LinearSeries( ['x^2', 'x + x^2'], ring )
assert str( ls.quo( 'x' ) ) == '{ 2, <<x, x + 1>>, QQ( <a0|t^2 + 1>, <a1|t^2 + a0*t - 1> )[x, y, z] }'
def test__aux_gcd(self):
ring = PolyRing('x,y,z', True)
ring.ext_num_field('t^2 + t + 1')
ring.ext_num_field('t^3 + t + a0 + 3')
agcd = ring.aux_gcd('[x*y^2 + a1*y^3, y^5, x^5*y^5]')
assert str(agcd) == '([(y, 2)], [x + a1*y, y^3, x^5*y^3])'
def test__diff( self ):
ring = PolyRing( 'x,y,z', True )
ring.ext_num_field( 't^2 + 1' )
ring.ext_num_field( 't^3 + a0' )
ls = LinearSeries( ['a0*x^2+y^3*z^3', 'x+z+z^3*y^4'], ring )
xls = ls.copy().chart( ['x'] )
assert str( xls.diff( 3, 2 ).pol_lst ) == '[36*z, 144*y*z]'
def test__diff__3(self):
ring = PolyRing('x,y,v,w', True)
ring.ext_num_field('t^2 + 1')
x, y, v, w, a0 = ring.coerce('x,y,v,w,a0')
d1 = ring.diff(y**2 * x**3 - a0, x, 2)
d2 = ring.diff(d1, y, 2)
assert d1 == 6 * y**2 * x
assert d2 == 12 * x
def test__chart( self ):
ring = PolyRing( 'x,y,z', True )
ring.ext_num_field( 't^2 + 1' )
ring.ext_num_field( 't^3 + a0' )
ls = LinearSeries( ['x^2+a0*y*z', 'x+y+a1*z '], ring )
xls = ls.copy().chart( ['x'] )
assert str( xls.pol_lst ) == '[a0*y*z + 1, y + a1*z + 1]'
def test__resultant(self):
ring = PolyRing('x,y,z', True)
ring.ext_num_field('t^2 + t + 1')
ring.ext_num_field('t^3 + t + a0 + 3')
pol1 = ring.coerce('(x+1)*(x^2+a0+1)')
pol2 = ring.coerce('x^5 + x + a0 + 3')
x = ring.coerce('x')
r = ring.resultant(pol1, pol2, x)
assert str(r) == '7*a0 + 2'
def test__quo(self):
ring = PolyRing('x,y,z', True)
ring.ext_num_field('t^2 + t + 1')
ring.ext_num_field('t^3 + t + a0 + 3')
pol1 = ring.coerce('(x+1)*(x^2+a0+1)')
pol2 = ring.coerce('(x+1)')
assert str(pol1) == 'x^3 + x^2 + (a0 + 1)*x + a0 + 1'
q = ring.quo(pol1, pol2)
assert str(q) == 'x^2 + a0 + 1'
def test__blow_up_origin( self ):
ring = PolyRing( 'x,y,z', True )
ring.ext_num_field( 't^2 + 1' )
ring.ext_num_field( 't^3 + a0' )
ls = LinearSeries( ['x^2', 'x*z + y^2'], ring )
zls = ls.copy().chart( ['z'] )
zls, m = zls.blow_up_origin( 's' )
assert str( zls ) == '{ 2, <<x, x*y^2 + 1>>, QQ( <a0|t^2 + 1>, <a1|t^2 + a0*t - 1> )[x, y] }'
assert m == 1
def test__factor(self):
ring = PolyRing('x,y,z', True)
ring.ext_num_field('t^2 + t + 1')
ring.ext_num_field('t^3 + t + a0 + 3')
pol = ring.coerce('(x+1)*(x^2+a0+1)')
assert str(pol) == 'x^3 + x^2 + (a0 + 1)*x + a0 + 1'
assert str(sage_factor(pol)) == '(x + 1) * (x + a0) * (x - a0)'
con = ring.coerce('a0')
assert str(sage_factor(con)) == 'a0'
def test__translate_to_origin( self ):
ring = PolyRing( 'x,y,z', True )
ring.ext_num_field( 't^2 + 1' )
ring.ext_num_field( 't^3 + a0' )
ls = LinearSeries( ['y^2', 'y*x'], ring )
xls = ls.copy().chart( ['x'] )
a0, a1 = ring.coerce( 'a0,a1' )
xls.translate_to_origin( ( a0 + 1, a1 + a0 ) )
assert str( xls ) == '{ 2, <<y^2 + (2*a0 + 2)*y + 2*a0, y + a0 + 1>>, QQ( <a0|t^2 + 1>, <a1|t^2 + a0*t - 1> )[y, z] }'
def test__get_base_point_tree(self):
ring = PolyRing('x,y,z', True)
ring.ext_num_field('t^2 + 1')
ring.ext_num_field('t^3 + a0')
ls = LinearSeries(['x^2+a0*y*z', 'y+a1*z+x'], ring)
out = str(ls.get_bp_tree())
chk = """
{ 2, <<x^2 + a0*y*z, x + y + a1*z>>, QQ( <a0|t^2 + 1>, <a1|t^2 + a0*t - 1>, <a2|t^2 - a0*t - a0*a1> )[x, y, z] }
chart=z, depth=0, mult=1, sol=(-a2 + a0, a2 - a1 - a0), { 2, <<x^2 + a0*y, x + y + a1>>, QQ( <a0|t^2 + 1>, <a1|t^2 + a0*t - 1>, <a2|t^2 - a0*t - a0*a1> )[x, y] }
chart=z, depth=0, mult=1, sol=(a2, -a2 - a1), { 2, <<x^2 + a0*y, x + y + a1>>, QQ( <a0|t^2 + 1>, <a1|t^2 + a0*t - 1>, <a2|t^2 - a0*t - a0*a1> )[x, y] }
"""
assert self.equal_output_strings(out, chk)
def test__get_linear_series__6(self):
ring = PolyRing('x,y,v,w', True)
ring.ext_num_field('t^2 + 1')
a0 = ring.root_gens()[0]
bp_tree_1 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bp = bp_tree_1.add('xv', (a0, -a0), 2)
bp_tree_2 = LinearSeries.get([2, 2], bp_tree_1).get_bp_tree()
bp_tree_2_str = bp_tree_2.alt_str()
bp_tree_2_str = bp_tree_2_str.replace('(a0)', 'a0')
bp_tree_2_str = bp_tree_2_str.replace('(-a0)', '-a0')
assert self.equal_output_strings(bp_tree_1.alt_str(), bp_tree_2_str)
def test__subs( self ):
ring = PolyRing( 'x,y,z', True )
ring.ext_num_field( 't^2 + 1' )
ring.ext_num_field( 't^3 + a0' )
a0, a1 = ring.root_gens()
x, y, z = ring.gens()
pol_lst = [x ** 2 + a0 * y * z, y + a1 * z + x ]
ls = LinearSeries( pol_lst, ring )
assert str( ls.pol_lst ) == '[x^2 + a0*y*z, x + y + a1*z]'
ls.subs( {x: x + 1} )
assert str( ls.pol_lst ) == '[x^2 + a0*y*z + 2*x + 1, x + y + a1*z + 1]'
def test__approx_QQ_coef__2(self):
ring = PolyRing('x,y,v,w', True)
ring.ext_num_field('t^2 - 3')
ring.ext_num_field('t^2 + 1')
ring.ext_num_field('t^2 + 2/7*a0*a1 + 3/7')
ring.ext_num_field('t^2 - 2/7*a0*a1 + 3/7')
a0, a1, a2, a3 = ring.root_gens()
q2 = sage_QQ(1) / 2
a = 2 * a0 / 3
b = (-a0 * a1 / 3 - q2) * a3
c = (a0 * a1 / 3 - q2) * a2
d = (a1 / 2 - a0 / 3) * a3
e = (-a1 / 2 - a0 / 3) * a2
ci_idx = 5 # index for complex embedding
C1 = -(b + c - d - e)
C2 = -(b + c + d + e)
for lbl, elt in [('a0', a0), ('C1', C1), ('C2', C2)]:
print(lbl + ' = ' + str(elt.abs()))
for ci in elt.complex_embeddings():
print('\t\t' + str(ci))
s = str(elt.complex_embeddings()[ci_idx])
print(lbl + '.complex_embedding()[' + str(ci_idx) + '] = ' + s)
# C1
print('--- C1 ---')
out = OrbRing.approx_QQ_coef(C1, ci_idx)
print(out)
print(sage_n(out))
chk = -sage_QQ(3687885631267691) / 2251799813685248
assert out == chk
# C2
print('--- C2 ---')
out = OrbRing.approx_QQ_coef(C2, ci_idx)
print(out)
print(sage_n(out))
chk = sage_QQ(2749869658604311) / 4503599627370496
assert out == chk
def test__ext_num_field(self):
ring = PolyRing('x,y,z', True)
assert str(ring) == 'QQ[x, y, z]'
ring.ext_num_field('t^2 + t + 1')
assert str(ring) == 'QQ( <a0|t^2 + t + 1> )[x, y, z]'
ring.ext_num_field('t^3 + t + a0 + 3')
assert str(
ring
) == 'QQ( <a0|t^2 + t + 1>, <a1|t^3 + t + a0 + 3>, <a2|t^2 + a1*t + a1^2 + 1> )[x, y, z]'
a = ring.root_gens()
x, y, z = ring.gens()
pol = x**3 + x + a[0] + 3
assert str(sage_factor(pol)) == '(x - a2) * (x - a1) * (x + a2 + a1)'
mat = list(pol.sylvester_matrix(y**2 + x**2, x))
assert str(
mat
) == '[(1, 0, 1, a0 + 3, 0), (0, 1, 0, 1, a0 + 3), (1, 0, y^2, 0, 0), (0, 1, 0, y^2, 0), (0, 0, 1, 0, y^2)]'
