4
x**3*y**2+4*x**2*y**2*z-x**2*y*z**2+288*x**2*y**2+207*x**2*y*z+1152*x*y**2*z+
156*x*y*z**2+x*z**3-3456*x**2*y+20736*x*y**2+19008*x*y*z+82944*y**2*z
+432*x*z**2- 497664*x*y+62208*x*z+2985984*x;
y**3*t**3+4*y**3*t**2-y**2*z*t**2+4*y**2*t**3-48*y**2*t**2-5*y*z*t**2
+108*y*z*t+ z**2*t+144*z*t-1728*z;
-x**2*z**2*t+4*x*z**2*t**2+z**3*t**2+x**3*z+156*x**2*z*t+207*x*z**2*t+
1152*x*z*t**2+288*z**2*t**2+432*x**2*z+19008*x*z*t-3456*z**2*t+82944*x*t**2+
20736*z*t**2+62208*x*z-497664*z*t+2985984*z;
y**3*t**3-x*y**2*t**2+4*y**3*t**2+4*y**2*t**3-5*x*y**2*t-48*y**2*t**2+x**2*y+
108*x*y*t+144*x*y-1728*x;
TITLE : cohn2, modular equations for special algebraic number fields
ROOT COUNTS :
total degree : 900
2-homogeneous Bezout number is 450
with partition : {x y z }{t }
generalized Bezout number : 358
based on the set structure :
{x }{x z }{x z }{y }{y z }
{y z }{y }{y z }{t }{t }{t }
{x z }{x t }{z }{x z }{t }
{y }{x y }{x y }{t }{t }{t }
mixed volume : 124
REFERENCES :
From PoSSo test suite.
Andre' Galligo and Carlo Traverso:
"Practical Determination of the dimension of an algebraic variety",
in E. Kaltofen and S.M. Watt, Eds "Computers and Mathematics",
pages 46-52, 1989.
H. Cohn: "An explicit modular equation in two variables and
Hilbert's Twelfth problem", Math. of Comp. 38, pp. 227-236, 1982.
H. Cohn, J. Deutch: "An explit modular equation in two variables
for Q[sqrt(3)]", Math. of Comp. 50, pp. 557-568, 1988.
THE SOLUTIONS :
18 4
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.23997158246700E+02 -7.54108290873468E+01
y : 4.81861750852218E+01 2.85294148295339E+01
z : -1.61588268408412E+01 1.78870081595856E+01
t : -3.05964654408600E+00 1.82602611365708E-01
== err : 2.401E-13 = rco : 2.653E-04 = res : 4.265E-06 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 6.02393786358094E+02 2.51345585423244E-88
y : 2.08383161251585E+01 -5.10545720390964E-89
z : 6.02393786358094E+02 -9.73964143515069E-88
t : 2.08383161251585E+01 -6.13636683162216E-89
== err : 5.985E-13 = rco : 5.275E-06 = res : 5.035E-04 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -5.97960427705686E+01 6.80828237761971E+00
y : -4.99290642112624E-01 9.33089973938555E+00
z : -5.97960427705686E+01 6.80828237761971E+00
t : -4.99290642112627E-01 9.33089973938555E+00
== err : 1.830E-14 = rco : 5.344E-04 = res : 2.107E-07 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -8.42105246557411E+02 -1.12909149701848E-89
y : -7.66022331457492E+00 8.43750439348047E-92
z : -1.09615038022715E+01 -3.68182009897330E-91
t : -2.18804311393714E+01 -2.87642195232289E-91
== err : 9.157E-14 = rco : 7.570E-07 = res : 9.507E-06 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -5.97960427705686E+01 -6.80828237761971E+00
y : -4.99290642112623E-01 -9.33089973938555E+00
z : -5.97960427705686E+01 -6.80828237761971E+00
t : -4.99290642112629E-01 -9.33089973938555E+00
== err : 2.214E-14 = rco : 5.344E-04 = res : 9.424E-08 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 2.30226349318897E+00 2.29757961177032E+02
y : 1.17605675035346E+00 -2.87963907718388E+00
z : 2.30226349318902E+00 -2.29757961177032E+02
t : 1.17605675035346E+00 2.87963907718388E+00
== err : 4.355E-13 = rco : 5.736E-07 = res : 3.279E-06 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.35414647301048E-01 -7.37888274251716E+00
y : 1.83176559758234E+00 -4.11273135653901E+00
z : 4.35414647301049E-01 -7.37888274251716E+00
t : 1.83176559758234E+00 -4.11273135653901E+00
== err : 1.640E-15 = rco : 2.781E-04 = res : 5.890E-09 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.35414647301048E-01 7.37888274251716E+00
y : 1.83176559758234E+00 4.11273135653901E+00
z : 4.35414647301049E-01 7.37888274251716E+00
t : 1.83176559758234E+00 4.11273135653901E+00
== err : 1.640E-15 = rco : 2.781E-04 = res : 5.890E-09 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 6.02648764055773E+02 -8.10922451376730E+02
y : 1.17711982346853E+00 9.71239630969622E-01
z : 6.02648764055773E+02 -8.10922451376730E+02
t : 1.17711982346853E+00 9.71239630969621E-01
== err : 4.823E-13 = rco : 1.265E-09 = res : 3.553E-04 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.09615038022715E+01 -4.29545678213551E-91
y : -2.18804311393714E+01 4.14204761134496E-91
z : -8.42105246557411E+02 1.62000084354825E-89
t : -7.66022331457492E+00 0.00000000000000E+00
== err : 1.783E-13 = rco : 7.580E-07 = res : 1.144E-05 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.61588268408412E+01 1.78870081595856E+01
y : -3.05964654408600E+00 1.82602611365708E-01
z : -1.23997158246700E+02 -7.54108290873469E+01
t : 4.81861750852218E+01 2.85294148295338E+01
== err : 2.167E-13 = rco : 3.095E-04 = res : 5.150E-06 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.36567442620473E+02 6.73357734978399E+01
y : -1.33747712198973E+01 -2.20687866697351E+00
z : 1.36567442620473E+02 6.73357734978399E+01
t : -1.33747712198973E+01 -2.20687866697351E+00
== err : 2.057E-13 = rco : 1.822E-05 = res : 2.861E-06 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.61588268408412E+01 -1.78870081595856E+01
y : -3.05964654408600E+00 -1.82602611365708E-01
z : -1.23997158246700E+02 7.54108290873469E+01
t : 4.81861750852218E+01 -2.85294148295338E+01
== err : 2.167E-13 = rco : 3.095E-04 = res : 5.150E-06 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 2.30226349318900E+00 -2.29757961177032E+02
y : 1.17605675035346E+00 2.87963907718388E+00
z : 2.30226349318901E+00 2.29757961177032E+02
t : 1.17605675035346E+00 -2.87963907718388E+00
== err : 3.868E-13 = rco : 5.736E-07 = res : 5.030E-06 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.36567442620473E+02 -6.73357734978399E+01
y : -1.33747712198973E+01 2.20687866697351E+00
z : 1.36567442620473E+02 -6.73357734978399E+01
t : -1.33747712198973E+01 2.20687866697351E+00
== err : 2.057E-13 = rco : 1.822E-05 = res : 2.861E-06 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -6.71049434640517E+01 7.06909459002873E-89
y : -1.21079632432404E+01 2.07409198908829E-89
z : -6.71049434640517E+01 -6.28363963558109E-89
t : -1.21079632432404E+01 -2.25818299403695E-89
== err : 5.000E-14 = rco : 3.370E-04 = res : 5.960E-08 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 6.02648764055773E+02 8.10922451376730E+02
y : 1.17711982346853E+00 -9.71239630969622E-01
z : 6.02648764055773E+02 8.10922451376730E+02
t : 1.17711982346853E+00 -9.71239630969621E-01
== err : 5.276E-13 = rco : 1.265E-09 = res : 3.200E-04 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.23997158246700E+02 7.54108290873468E+01
y : 4.81861750852218E+01 -2.85294148295339E+01
z : -1.61588268408412E+01 -1.78870081595856E+01
t : -3.05964654408600E+00 -1.82602611365708E-01
== err : 2.401E-13 = rco : 2.653E-04 = res : 4.265E-06 ==