Cassou Nogues

Equations:

15b^{4}cd^{2} + 6b^{4}c^{3} + 21b^{4}c^{2}d − 144b^{2}c − 8b^{2}c^{2}e − 28b^{2}cde − 648b^{2}d + 36b^{2}d^{2}e +

9b^{4}d^{3} − 120; 30c^{3}b^{4}d − 32de^{2}c − 720db^{2}c − 24c^{3}b^{2}e − 432c^{2}b^{2} + 576ec − 576de + 16cb^{2}d^{2}e +

16d^{2}e^{2} +16e^{2}c^{2} +9c^{4}b^{4} +5184+39d^{2}b^{4}c^{2} +18d^{3}b^{4}c−432d^{2}b^{2} +24d^{3}b^{2}e− 16c^{2}b^{2}de − 240c;

216db^{2}c − 162d^{2}b^{2} − 81c^{2}b^{2} + 5184 + 1008ec − 1008de + 15c^{2}b^{2}de − 15c^{3}b^{2}e −

80de^{2}c + 40d^{2}e^{2} + 40e^{2}c^{2}; 261 + 4db^{2}c − 3d^{2}b^{2} − 4c^{2}b^{2} + 22ec − 22de

Communicated by Pierrette Cassou-Nogués to the PoSSo tests suite.

Characteristics:

• Dimension 0.

• Degree 16. Number of different complex solutions: 16. Number of different real solutions: 4.

• A plane (dimension 2) at infinity.

• The Generalized Shape Lemma description for this system is

{(β, R_{c}(β), R_{d}(β), R_{e}(β)) ∈ C^{4} : E(β) = 0}

where

R_{⋆}(T) = \frac{F_{⋆}(T)}{E′(T)} ⋆ ∈ \{c, d, e\}

and

E = T^{16} − \frac{11328065425280}{5581434681}T^{14} − \frac{1982959945089290240}{4068865882449}T^{12} − \frac{38925480508049063936}{8898609684915963}T^{10} + \frac{145090425457775476736}{6487086460303737027}T^{8}

+\frac{3121544456059492499456}{14187258088684272878049}T^{6} + \frac{5952612054194978816}{3447503715550278309365907}T^{4} + \frac{110637258033332224}{22619071877725375987749715827}T^{2}

+\frac{281474976710656}{49467910196585397285208628513649}

F_{c} = \frac{7533}{2}T^{15} − \frac{526748459471840}{68906601}T^{13} − \frac{92179019614471057408}{50232912129}T^{11} \frac{1764804584247569416192}{109859378826123}T^{9} + \frac{2192692928086374612992}{26695829054747889}T^{7}

+\frac{133396983688410544734208}{175151334428200899729}T^{5} + \frac{388693091715790143488}{42561774266052818634147}T^{3} + \frac{9318587544796921856}{279247800959572543058638467}T

F_{d} = 15876T^{15} − \frac{6660873508296224}{206719803}T^{13} − \frac{388674056815354689536}{50232912129}T^{11} − \frac{22910176488147242713088}{329578136478369}T^{9}

+\frac{83818914488883903201280}{240262461492731001}T^{7} + \frac{1818531842440503027040256}{525454003284602699187}T^{5} + \frac{10763839267814298877952}{383055968394475367707323}T^{3}

+\frac{21010306011404173312}{279247800959572543058638467}T

F_{e} = −\frac{3826500880}{10384551}T^{15} + \frac{729760960838961152}{6982374785931}T^{13} − \frac{31884427708271230189568}{45811360970493291}T^{11} − \frac{146275085402355093471232}{11132160715829869713}T^{9}

−\frac{4649723553000264814297088}{73038106456559775186993}T^{7} + \frac{34442087585624908365824}{53244779606832076111317897}T^{5}

−\frac{17184203458836692992}{12938481444460194495050248971}T^{3} + \frac{371476600513888256}{84889376757103336082024683498731}T

Example 1:

vars:= [b,c,d,e];
[15*b**4*c*d**2+6*b**4*c**3+21*b**4*c**2*d-144*b**2*c-8*b**2*c**2*e
-28*b**2*c*d*e-648*b**2*d+36*b**2*d**2*e+9*b**4*d**3-120,
30*c**3*b**4*d-32*d*e**2*c-720*d*b**2*c-24*c**3*b**2*e-432*c**2*b**2
+576*e*c-576*d*e+16*c*b**2*d**2*e+16*d**2*e**2+16*e**2*c**2+9*c**4*b**4+5184
+39*d**2*b**4*c**2+18*d**3*b**4*c-432*d**2*b**2+24*d**3*b**2*e-16*c**2*b**2*d*e
-240*c,
216*d*b**2*c-162*d**2*b**2-81*c**2*b**2+5184+1008*e*c-1008*d*e
+15*c**2*b**2*d*e-15*c**3*b**2*e-80*d*e**2*c+40*d**2*e**2+40*e**2*c**2,
261+4*d*b**2*c-3*d**2*b**2-4*c**2*b**2+22*e*c-22*d*e];

Problem: Find the roots of the systems.

References