```
6
12*H1 - 0.51234*H1*(1+(1/2)*H1+(1/3)*H2+(1/4)*H3+(1/5)*H4+(1/6)*H5) - 12;
12*H2 - 0.51234*H2*(1+(2/3)*H1+(2/4)*H2+(2/5)*H3+(2/6)*H4+(2/7)*H5) - 12;
12*H3 - 0.51234*H3*(1+(3/4)*H1+(3/5)*H2+(3/6)*H3+(3/7)*H4+(3/8)*H5) - 12;
12*H4 - 0.51234*H4*(1+(4/5)*H1+(4/6)*H2+(4/7)*H3+(4/8)*H4+(4/9)*H5) - 12;
12*H5 - 0.51234*H5*(1+(5/6)*H1+(5/7)*H2+(5/8)*H3+(5/9)*H4+(5/10)*H5) - 12;
12*H6 - 0.51234*H6*(1+(6/7)*H1+(6/8)*H2+(6/9)*H3+(6/10)*H4+(6/11)*H5) - 12;

TITLE : The Chandrasekhar H-Equation for n=6

ROOT COUNTS :

total degree : 64
2-homogeneous Bezout number : 32
with partition : {H1 H2 H3 H4 H5 }{H6 }
generalized Bezout number : 32
based on the set structure :
{H1 }{H1 H2 H3 H4 H5 }
{H1 H2 }{H2 H3 H4 H5 }
{H1 H2 H3 }{H3 H4 H5 }
{H1 H2 H3 H4 }{H4 H5 }
{H1 H2 H3 H4 H5 }{H5 }
{H1 H2 H3 H4 H5 }{H6 }
mixed volume : 32

REFERENCES :

Laureano Gonzalez-Vega: "Some examples on problem solving by using the
symbolic viewpoint when dealing with polynomial systems of equations".
in: "Computer Algebra in Science and Engineering", Editors: J. Fleischer,
J. Grabmeier, F.W. Hehl and W. Kuechlin.  Pages 102-116
World Scientific Publishing, 1995.

S. Chandrasekhar: "Radiative Transfer", Dover, NY, 1960.

C.T. Kelley: "Solution of the Chandrasekhar H-equation by Newton's method".
J. Math. Phys., 21 (1980), pp. 1625-1628.

Jorge J. More': "A collection of nonlinear model problems"
in: "Computational Solution of Nonlinear Systems of Equations",
Editors: Eugene L. Allgower and Kurt Georg.  Pages 723-762.
Lectures in Applied Mathematics, Volume 26, AMS, 1990.

NOTE : the parameter c equals 0.51234.
In general c can be any number in the interval (0,1]

TIMING : black-box solver on juno, executed on 9/7/1998.
---------------------------------------------------------------------
|                    TIMING INFORMATION SUMMARY                     |
---------------------------------------------------------------------
|   root counts  |  start system  |  continuation  |   total time   |
---------------------------------------------------------------------
|  0h 0m 1s964ms |  0h 0m 0s113ms |  0h 2m10s594ms |  0h 2m14s755ms |
---------------------------------------------------------------------

THE SOLUTIONS :
32 6
===========================================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  3.52478437068955E+00  -6.43803527809232E-106
H2 :  1.46544365626337E+01  -5.01928305940161E-105
H3 :  2.76720166523013E+02   1.78766185922902E-104
H4 : -3.57186149421082E+02  -9.06911382243016E-104
H5 :  6.83219998395596E+01   8.91650854176427E-104
H6 :  2.40247645670325E+01   9.15086665135832E-104
== err :  1.469E-11 = rco :  3.164E-04 = res :  1.029E-12 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  2.18039077320382E+00   5.68786937493494E-121
H2 :  5.26610247476354E+00  -3.38851792549315E-121
H3 :  4.75105406270813E+01   1.12305165530630E-119
H4 : -1.05582510453516E+01  -1.31668125104877E-119
H5 : -5.59012797702533E+00  -5.80888787227398E-121
H6 : -4.11914439626348E+00   1.74266636168219E-120
== err :  5.308E-14 = rco :  2.807E-02 = res :  2.864E-14 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  7.06422178828185E+01   1.62145877294106E-121
H2 : -8.00514340830750E+00  -1.13454841255351E-121
H3 : -8.38935162117220E+00  -1.96655058175942E-121
H4 : -1.28876016124331E+01  -6.53499885630822E-121
H5 : -3.53248833660917E+01   8.83435030575001E-121
H6 :  4.99495886280820E+01  -2.17639665614532E-118
== err :  3.934E-12 = rco :  3.095E-03 = res :  2.136E-13 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  4.58804372895969E+00   6.81576177013480E-118
H2 :  1.04960253120326E+02   4.86273621563799E-116
H3 : -2.28718536317627E+01   1.16983256563768E-116
H4 : -2.41697569481786E+01  -2.50671325611430E-115
H5 : -5.64714483945308E+01   1.90742191210754E-115
H6 :  7.28492743509224E+01   9.13659061433634E-115
== err :  2.831E-12 = rco :  1.277E-03 = res :  1.967E-13 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.38740582816837E+02   4.20689238953442E-112
H2 : -1.58482351149297E+02  -1.31242048291267E-111
H3 :  1.42464797385588E+01   9.74569665529209E-112
H4 :  6.80738141178888E+00  -6.11136394425608E-113
H5 :  4.72314505692644E+00  -2.20293351478998E-113
H6 :  3.77476995479413E+00  -1.42949573986804E-113
== err :  8.823E-13 = rco :  4.484E-03 = res :  5.480E-13 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  3.13287093598037E+00   8.10841961720302E-111
H2 :  1.11527393390388E+01  -1.49693900625287E-110
H3 :  1.58071147610527E+02   1.99591867500382E-110
H4 : -6.40738501585373E+01   3.69244954875707E-109
H5 : -1.02247669852195E+02  -4.25795984000815E-109
H6 :  1.13108398486882E+02  -2.47387466704474E-106
== err :  6.466E-12 = rco :  4.973E-04 = res :  5.711E-13 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  5.06454835814171E+00   7.04124123745707E-94
H2 :  3.10356875748233E+01   1.21079386555592E-91
H3 :  9.20668839208376E+02  -5.99554200687302E-91
H4 : -2.16762148564448E+03   1.03952690847802E-90
H5 :  1.24966106535584E+03  -5.90115941156242E-91
H6 : -6.59700749166740E+02  -4.16948944381597E-86
== err :  8.999E-09 = rco :  7.303E-06 = res :  5.323E-11 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.53149889674569E+02   3.25168417469372E-109
H2 : -4.68616318097814E+01  -7.91714407751516E-109
H3 : -5.73468603588809E+02   1.77054619128464E-108
H4 :  6.10507844195193E+02  -6.11083767663670E-108
H5 : -1.04518843618498E+02   5.28918448876012E-108
H6 : -3.41831622806402E+01   2.02120031155387E-107
== err :  2.461E-10 = rco :  9.160E-05 = res :  9.673E-12 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  6.15292600621423E+00   3.82551079375732E-110
H2 :  2.20797087828788E+02  -2.96393923238067E-108
H3 : -2.68430446519935E+02   8.56249111576639E-108
H4 :  3.38445546089952E+01  -6.50669488051246E-108
H5 :  1.36711159507517E+01   7.11877660751363E-109
H6 :  8.54778179300395E+00  -5.32244980001019E-110
== err :  3.234E-12 = rco :  9.179E-04 = res :  5.493E-13 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.14199064022975E+02  -7.60034396490713E-110
H2 : -2.22766014930415E+01   2.28699014844188E-110
H3 : -4.88628996216393E+01   8.43587502482084E-109
H4 : -4.35988193067735E+02  -1.33809714503381E-108
H5 :  4.31737285012113E+02   5.26839366922883E-109
H6 : -2.91329215867664E+02  -4.56759867957354E-104
== err :  2.200E-10 = rco :  6.809E-05 = res :  5.732E-12 ==
solution 11 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  9.94672041693105E+00   1.09521740091832E-120
H2 :  6.14431058322047E+02  -1.29013281819502E-119
H3 : -1.56344501410518E+03   3.12711797124082E-119
H4 :  1.14496293824953E+03  -2.23581673833879E-119
H5 : -1.67087048030661E+02   2.31674785843427E-120
H6 : -4.98546361554169E+01  -9.60886868871987E-119
== err :  5.018E-10 = rco :  1.819E-05 = res :  3.819E-11 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  2.20051765081231E+02  -3.72238612550373E-122
H2 : -9.92452325191326E+01   2.91658714961510E-121
H3 : -1.90797324323279E+03  -7.50252637885744E-121
H4 :  3.70493067096193E+03   8.03478913021555E-121
H5 : -1.91172872241648E+03  -3.11572403244350E-121
H6 :  9.38642195702171E+02  -1.53777720494730E-116
== err :  5.760E-08 = rco :  2.804E-06 = res :  9.682E-11 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  3.84126257293832E+02  -3.48117171047578E-103
H2 : -1.49531906105105E+03   2.76793277996357E-102
H3 :  1.85081599242891E+03  -6.66318256896047E-102
H4 : -3.51054285332415E+02   6.17876980661816E-102
H5 : -3.82533665464453E+02  -1.91061811393697E-102
H6 :  3.33959929087173E+02   7.40144331531329E-101
== err :  3.093E-09 = rco :  1.246E-05 = res :  5.007E-11 ==
solution 14 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  9.47367473427292E+01   7.26110984034247E-120
H2 : -1.68398255497713E+01   1.54903676593973E-120
H3 : -9.84597679724875E+01  -4.33730294463124E-119
H4 :  1.80463186901779E+01   1.60712564466247E-119
H5 :  8.55176536416566E+00   2.16865147231562E-119
H6 :  5.86084333779961E+00  -8.28976706772432E-121
== err :  3.160E-13 = rco :  5.725E-03 = res :  1.856E-13 ==
solution 15 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  2.67341159040361E+02  -3.46784372650810E-112
H2 : -7.06060024230957E+02   2.17329035413014E-111
H3 :  5.56289175670613E+02  -3.37525960828283E-111
H4 : -5.78476128079951E+01   9.30714030580395E-112
H5 : -2.09140428193520E+01   6.46464544801042E-112
H6 : -1.21620426853564E+01   1.71361832855553E-112
== err :  8.068E-11 = rco :  1.352E-04 = res :  5.530E-12 ==
solution 16 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  5.16199523004502E+00  -1.01517673492626E-115
H2 :  1.37915298133592E+02   2.43642416382302E-114
H3 : -4.00395849675597E+01  -4.87284832764605E-114
H4 : -1.34736751348682E+02   2.37145285278774E-112
H5 :  3.77342808274185E+01  -2.69630940796415E-112
H6 :  1.54735341369303E+01   8.60869871217468E-113
== err :  1.421E-12 = rco :  1.556E-03 = res :  5.849E-13 ==
solution 17 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  3.22262596020634E+02   1.95130807439710E-94
H2 : -9.34013107408185E+02  -1.67083177003521E-93
H3 :  2.76071157906576E+02   4.46456361374335E-93
H4 :  1.39756349283305E+03  -4.72673748960611E-93
H5 : -1.05584890147724E+03   1.73162041145317E-93
H6 :  6.04547529200941E+02  -1.40136880498877E-89
== err :  4.122E-09 = rco :  8.591E-06 = res :  2.737E-11 ==
solution 18 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.13154550436088E+00  -4.84073989356165E-122
H2 :  1.18203024240421E+00   0.00000000000000E+00
H3 :  1.21673759622537E+00  -4.84073989356165E-122
H4 :  1.24247204712991E+00  -9.68147978712330E-122
H5 :  1.26245248469364E+00  -1.93629595742466E-121
H6 :  1.27847129867409E+00  -1.45222196806849E-121
== err :  4.215E-16 = rco :  6.287E-01 = res :  8.049E-16 ==
solution 19 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  2.24285776615435E+02   5.68781681796161E-103
H2 : -4.41022477694693E+02  -2.50970284432250E-102
H3 :  8.29771287294896E+01   1.36734331476639E-102
H4 :  2.30294046208341E+02   3.65729455493001E-102
H5 : -5.77258190059000E+01  -3.11837990663560E-102
H6 : -2.20162127700374E+01   1.74406035046734E-104
== err :  1.364E-11 = rco :  2.901E-04 = res :  2.507E-12 ==
solution 20 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.82923811525524E+00   2.71081434039452E-120
H2 :  3.28933728225724E+00   5.80888787227398E-121
H3 :  7.08676065981003E+00   8.42288741479727E-120
H4 :  4.20328901416938E+01   3.71768823825535E-119
H5 : -1.54295713463446E+01  -5.26672500419507E-119
H6 : -7.45663880688855E+00   6.97066544672877E-120
== err :  1.483E-14 = rco :  1.865E-02 = res :  1.066E-14 ==
solution 21 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  7.94793627290892E+01  -4.38556349488144E-113
H2 : -1.05185697150827E+01  -2.92370899658763E-113
H3 : -1.46864422301210E+01   1.07202663208213E-112
H4 : -7.18432368872510E+01   4.02822128418740E-112
H5 :  2.36041239781795E+01  -4.93781963868133E-112
H6 :  1.06095314146729E+01  -1.46185449829381E-113
== err :  6.239E-13 = rco :  5.664E-03 = res :  3.091E-13 ==
solution 22 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  6.20972279252226E+02   7.35387919477345E-107
H2 : -4.16115303994330E+03  -3.46256961536350E-106
H3 :  1.07798838506612E+04   7.70088628907849E-106
H4 : -1.18761836483198E+04  -1.07047438949592E-105
H5 :  4.67528921320353E+03   5.88249626388845E-106
H6 : -1.94780951950533E+03  -8.88420435832877E-98
== err :  9.365E-07 = rco :  3.252E-07 = res :  5.081E-09 ==
solution 23 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  4.91650129283578E+01  -6.92473396333930E-128
H2 : -3.77987005988417E+00   1.15412232722322E-127
H3 : -2.52153942738901E+00   8.65591745417413E-129
H4 : -2.12365158110310E+00   8.65591745417413E-129
H5 : -1.93129700730984E+00   6.60363175579126E-129
H6 : -1.81904766441278E+00   3.89641092067872E-129
== err :  1.198E-14 = rco :  1.490E-01 = res :  5.596E-14 ==
solution 24 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.62584893814356E+00  -3.46236698166965E-128
H2 :  2.50334573768182E+00   4.61648930889287E-128
H3 :  4.04817763884991E+00  -4.61648930889287E-128
H4 :  7.54007149782880E+00   1.78888960719599E-127
H5 :  2.30912110401676E+01  -6.92473396333930E-128
H6 : -3.51057955714387E+01   1.84659572355715E-126
== err :  2.290E-14 = rco :  8.167E-03 = res :  1.305E-14 ==
solution 25 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.99347908131701E+02  -8.17528289281565E-105
H2 : -3.35639566582015E+02   6.01700820911232E-104
H3 :  4.73991112700689E+01   5.31938406892538E-104
H4 :  4.13113056961279E+01  -4.58687872172910E-103
H5 :  8.63898963367872E+01   3.74972975350478E-103
H6 : -1.03652152770022E+02   5.42053956925249E-102
== err :  1.898E-11 = rco :  3.596E-04 = res :  2.442E-12 ==
solution 26 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  8.84076528528998E+00  -5.15745385620987E-107
H2 :  4.67611934855825E+02  -8.86134667203696E-105
H3 : -8.93088316297117E+02   3.83842305697215E-104
H4 :  2.05389217530589E+02  -5.14642574022425E-104
H5 :  2.50055053478088E+02   2.19267899921060E-104
H6 : -2.34715226321462E+02   1.23910037712109E-103
== err :  6.830E-10 = rco :  4.683E-05 = res :  1.192E-11 ==
solution 27 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  2.62832103157822E+00   3.51281686800672E-108
H2 :  6.96627562471526E+00   1.71915128540329E-107
H3 :  2.35781865571705E+01   1.11771445800214E-106
H4 :  2.55080987665569E+02  -3.50855890816672E-106
H5 : -2.82218533004220E+02   2.14601175936411E-106
H6 :  2.04753315835934E+02   7.84478345640209E-102
== err :  2.009E-10 = rco :  1.664E-04 = res :  1.785E-12 ==
solution 28 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  3.19315044191211E+00   4.84073989356165E-122
H2 :  4.95601512692032E+01   2.42036994678082E-122
H3 : -6.87446221283792E+00   0.00000000000000E+00
H4 : -3.98275366522494E+00  -1.93629595742466E-121
H5 : -3.08743098038080E+00  -1.49382207652879E-122
H6 : -2.65300087552096E+00   2.42036994678082E-122
== err :  1.542E-14 = rco :  5.413E-02 = res :  1.377E-14 ==
solution 29 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  4.32179382984064E+02  -2.21505468443320E-143
H2 : -1.96481399366718E+03   1.06490799832789E-142
H3 :  3.24004802502049E+03   2.26235389755690E-143
H4 : -1.95698757145588E+03  -4.22759866378864E-142
H5 :  2.55609394993315E+02   3.26880106950545E-142
H6 :  7.09346854735957E+01  -1.32818192534447E-142
== err :  1.896E-08 = rco :  4.075E-06 = res :  1.257E-10 ==
solution 30 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.42918378144617E+01   4.18208726722381E-119
H2 :  1.30126397429125E+03  -2.25297347397912E-118
H3 : -5.20169933548546E+03   6.36643521682711E-118
H4 :  6.94832727772565E+03   1.00421527274741E-118
H5 : -3.05614851647160E+03  -8.00674808686401E-118
H6 :  1.36896828770925E+03  -2.34266783371164E-111
== err :  5.440E-07 = rco :  9.123E-07 = res :  9.696E-10 ==
solution 31 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  7.41695710081988E+00   6.30451306976616E-99
H2 :  2.92081929343525E+02   3.61032212632294E-97
H3 : -1.33214715348052E+02  -1.57416509730112E-96
H4 : -8.17664060049490E+02   2.09853913773665E-96
H5 :  6.90188543805867E+02  -8.88915581542616E-97
H6 : -4.24890826053001E+02   1.33263917436313E-92
== err :  1.233E-09 = rco :  2.472E-05 = res :  1.854E-11 ==
solution 32 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
H1 :  1.36121472337384E+02   1.46042690601032E-113
H2 : -3.56639821901531E+01  -1.18252227482426E-113
H3 : -3.27583064967396E+02  -1.79813179173814E-112
H4 :  1.09515971414293E+02   3.58052834408492E-112
H5 :  1.56418258258544E+02  -1.80016214520799E-112
H6 : -1.60934053276268E+02   1.30429906903326E-111
== err :  6.656E-11 = rco :  1.766E-04 = res :  1.643E-12 ==
```