5
y1*y2 + y1 - 3*y5;
2*y1*y2 + y1 + 1.9230E-06*y2**2 + y2*y3**2 + 5.4518E-04*y2*y3
+ 3.4074E-05*y2*y4 + 4.4975E-07*y2 - 10*y5;
2*y2*y3**2 + 5.4518E-04*y2*y3 + 3.8600E-01*y3**2 + 4.1062E-04*y3 - 8*y5;
3.4074E-05*y2*y4 + 2*y4**2 - 40*y5;
y1*y2 + y1 + 9.6150E-07*y2**2 + y2*y3**2 + 5.4518E-04*y2*y3 + 3.4074E-05*y2*y4
+ 4.4975E-07*y2 + 1.930E-01*y3**2 + 4.1062E-04*y3 + y4**2 - 1;
TITLE : chemical equilibrium of hydrocarbon combustion
ROOT COUNTS :
total degree : 108
3-homogeneous Bezout number : 56
with partition {y1 }{y2 y5 y4 }{y3 }
generalized Bezout bound is 44,
based on
{y1 y5 }{y2 }
{y1 y2 y5 }{y2 y3 y4 }{y3 }
{y2 y5 }{y3 }{y3 }
{y2 y5 y4 }{y4 }
{y1 y2 y4 }{y2 y3 y4 }{y3 }
mixed volume : 16
REFERENCES :
This polynomial system describes the equilibrium of the products
of hydrocarbon combustion.
Keith Meintjes and Alexander P. Morgan:
"Chemical equilibrium systems as numerical test problems",
ACM Toms, Vol 16, No 2, 143-151, 1990.
NOTES :
Although the total degree equals 108, there are only 4 real and
12 complex solutions and an infinite number of solutions at infinity.
A typographical error has occured in equation (2d),
instead of `+ 4Ry5', it should be a `- 4Ry5'.
Applying m-homogenization straight to it renders B = 56.
Simple linear reduction makes the total degree equal to 48.
With m-homogenization, no better upper bound can then be computed.
The constants are :
R = 10;
p = 40;
sqrt(p) = 6.3246
1/sqrt(p) = 0.1581
1/p = 0.0250
R5 = 1.930E-01 (2*R5 = 3.8600E-01)
R6 = 2.597E-03/sqrt(p) = 4.1062E-04
R7 = 3.448E-03/sqrt(p) = 5.4518E-04
R8 = 1.799E-05/p = 4.4975E-07
R9 = 2.155E-04/sqrt(p) = 3.4074E-05
R10 = 3.846E-05/p = 9.6150E-07
(2*R10 = 1.9230E-06)
THE SOLUTIONS :
16 5
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.38630037350383E-03 -1.91542055776285E-03
y2 : -2.86049955834289E+01 3.80706213979012E+01
y5 : 3.70633555200751E-02 3.26197913397758E-05
y3 : 2.49295017279909E-02 -4.99248345039094E-02
y4 : 8.61212433038939E-01 5.44767381799444E-05
== err : 3.011E-13 = rco : 5.147E-07 = res : 8.576E-17 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 1.73113619227222E-01 1.00759753038847E-04
y2 : -3.58157473618939E-01 -3.08119934353537E-04
y5 : 3.70372379206193E-02 3.77737915199671E-06
y3 : -5.09180689373301E-04 -9.47109079516830E-01
y4 : 8.60668351965905E-01 4.38916852917806E-05
== err : 4.344E-16 = rco : 8.635E-03 = res : 1.110E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.22212992907913E-03 -2.80225128764545E-03
y2 : -1.56097422221056E+01 3.33169447718268E+01
y5 : 3.70724848710101E-02 7.41778676457834E-05
y3 : -3.44166565964789E-02 5.35200064242835E-02
y4 : -8.60942156715559E-01 -1.14522296229995E-03
== err : 1.696E-13 = rco : 6.970E-07 = res : 8.066E-17 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.57729760800450E-03 -2.22202282076660E-03
y2 : -2.46854897571808E+01 3.32492893427509E+01
y5 : 3.70799153440935E-02 6.18914044395885E-05
y3 : -2.72118749521147E-02 5.34657885776007E-02
y4 : 8.61371584905004E-01 4.35393767521562E-04
== err : 3.503E-13 = rco : 6.000E-07 = res : 1.164E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 2.75718040490632E-03 9.65745061263782E-53
y2 : 3.92422451862829E+01 -4.91788706656964E-49
y5 : 3.69850432923516E-02 3.26265223399926E-54
y3 : -6.13876389840001E-02 -3.54976563059120E-52
y4 : 8.59724420833890E-01 0.00000000000000E+00
== err : 1.040E-13 = rco : 1.092E-06 = res : 8.544E-17 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 2.15330286265060E-03 1.01559609334694E-57
y2 : 5.05496866626878E+01 -1.10930175955975E-53
y5 : 3.70006959531691E-02 1.20726496390506E-58
y3 : -5.41447465741199E-02 -1.24261169068331E-56
y4 : -8.60671332237121E-01 -1.29812098696183E-57
== err : 1.156E-13 = rco : 6.918E-07 = res : 5.551E-17 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 3.11410764809440E-03 -1.01957882312477E-56
y2 : 3.45978628309741E+01 3.42447978480563E-51
y5 : 3.69518589659172E-02 -1.78426294046835E-56
y3 : 6.50418355152163E-02 -1.30506089359970E-54
y4 : 8.59378045022829E-01 -2.38326549905415E-55
== err : 1.053E-13 = rco : 1.352E-06 = res : 1.110E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.22212992907914E-03 2.80225128764545E-03
y2 : -1.56097422221056E+01 -3.33169447718267E+01
y5 : 3.70724848710101E-02 -7.41778676457844E-05
y3 : -3.44166565964790E-02 -5.35200064242836E-02
y4 : -8.60942156715559E-01 1.14522296229997E-03
== err : 3.128E-13 = rco : 6.970E-07 = res : 5.606E-17 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 2.47099675027988E-03 -2.61375006529336E-46
y2 : 4.38792820192678E+01 1.88334513605255E-42
y5 : 3.69655200081684E-02 -4.44748047759341E-48
y3 : 5.77844154385579E-02 -3.34997914127652E-45
y4 : -8.60205478426535E-01 2.73691106313441E-47
== err : 2.720E-13 = rco : 8.217E-07 = res : 2.821E-16 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.57729760800454E-03 2.22202282076661E-03
y2 : -2.46854897571809E+01 -3.32492893427503E+01
y5 : 3.70799153440935E-02 -6.18914044395882E-05
y3 : -2.72118749521145E-02 -5.34657885776011E-02
y4 : 8.61371584905004E-01 -4.35393767521564E-04
== err : 1.414E-14 = rco : 6.000E-07 = res : 2.068E-16 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 1.73116100360128E-01 -1.00769456612651E-04
y2 : -3.58173414218133E-01 3.08154503561680E-04
y5 : 3.70368488968581E-02 -3.77667010792754E-06
y3 : -5.09203021995388E-04 9.47058402634478E-01
y4 : -8.60657729853578E-01 4.38784276875199E-05
== err : 5.643E-16 = rco : 8.634E-03 = res : 9.714E-17 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.05873332033354E-03 -2.45884369206460E-03
y2 : -1.74700517946024E+01 3.81184217502881E+01
y5 : 3.70548778315729E-02 4.66799125262148E-05
y3 : 3.21415658535543E-02 -4.99961359811686E-02
y4 : -8.60721535891716E-01 -8.66896553728644E-04
== err : 2.351E-13 = rco : 5.781E-07 = res : 1.141E-16 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.38630037350383E-03 1.91542055776285E-03
y2 : -2.86049955834289E+01 -3.80706213979012E+01
y5 : 3.70633555200751E-02 -3.26197913397761E-05
y3 : 2.49295017279909E-02 4.99248345039094E-02
y4 : 8.61212433038939E-01 -5.44767381799484E-05
== err : 2.778E-13 = rco : 5.147E-07 = res : 8.565E-17 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 1.73116100360128E-01 1.00769456612651E-04
y2 : -3.58173414218133E-01 -3.08154503561680E-04
y5 : 3.70368488968581E-02 3.77667010792754E-06
y3 : -5.09203021995389E-04 -9.47058402634478E-01
y4 : -8.60657729853578E-01 -4.38784276875199E-05
== err : 6.410E-16 = rco : 8.634E-03 = res : 1.110E-16 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : 1.73113619227222E-01 -1.00759753038847E-04
y2 : -3.58157473618939E-01 3.08119934353537E-04
y5 : 3.70372379206193E-02 -3.77737915199671E-06
y3 : -5.09180689373301E-04 9.47109079516830E-01
y4 : 8.60668351965905E-01 -4.38916852917806E-05
== err : 4.344E-16 = rco : 8.635E-03 = res : 1.110E-16 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
y1 : -1.05873332033354E-03 2.45884369206460E-03
y2 : -1.74700517946023E+01 -3.81184217502881E+01
y5 : 3.70548778315729E-02 -4.66799125262152E-05
y3 : 3.21415658535544E-02 4.99961359811686E-02
y4 : -8.60721535891716E-01 8.66896553728648E-04
== err : 2.423E-13 = rco : 5.781E-07 = res : 1.123E-16 ==