# f = x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

# L, the resolvent
t^24 - 16926*t^23 - 379411449*t^22 + 16029851579299*t^21 + 128832700475364051*t^20 - 4487273414795234877871*t^19 - 12259111554208145022325329*t^18 + 1247499915926573402772636097579*t^17 + 16169730205832065501291099383849321*t^16 + 110983446780082443454615532790364910204*t^15 + 854860170188613598740098564533273670247006*t^14 + 5993333057259631153319469802151490579724327444*t^13 + 27718131097079425248226270884294625691629544991431*t^12 + 88493685731231900207602114227513195076547876172889669*t^11 + 47880531660121618604112799784984369678015405039558898806*t^10 - 1170778250882284528527739257825098591645649610969740301969246*t^9 - 2353561364270349956334668953936351695009109306058118952761805579*t^8 + 8164714007134271555131471271633498480758008540384148238973902206829*t^7 + 10450048105427463119142541888108981909126091710617931080429919593123571*t^6 - 35438875982449629664933373487962267936907229595066976459505974162275554921*t^5 + 24130152087129237846818376564356302530538936305514975176809923645078059152751*t^4 - 1334350552004200777487097812039020366406619810903658389049695736480985274157051*t^3 + 178475118340427767282446026180973922064599870587569309312443381009012542730908801*t^2 - 30042774775650318872630366999851978768719125288697753380858709406279024790703367951*t + 1644253690711002526116398825272052452909413015465395778937516374770941782706538370551

# psis: -1 and the roots of L
[-1.00000000000000, -5335.94085320292 - 396.088640483516*I, -1003.55914679708 + 5255.66551816988*I, -5335.94085320292 + 396.088640483517*I, -1003.55914679708 - 5255.66551816988*I]

# purported root r:
1/5*(1/2*(1/2)^(1/5)*(5115*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 3720*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 + 34720*sqrt(5) + 34720*I*sqrt(2*sqrt(5) + 10) - 106144)^(1/5)) + 1/5*(-465/32*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 155/2*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 + 4185/4*sqrt(5) + 4185/4*I*sqrt(2*sqrt(5) + 10) - 12369/4)^(1/5) + 1/5*(-155/8*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 2945/16*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 - 1085/2*sqrt(5) - 1085/2*I*sqrt(2*sqrt(5) + 10) - 5487/2)^(1/5) + 1/5*(-2945/64*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 - 5115/16*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 - 4185/4*sqrt(5) - 4185/4*I*sqrt(2*sqrt(5) + 10) - 20739/4)^(1/5) - 1/5

# f(r):  2.27373675443232e-13 + 2.23304628921836e-14*I, which is basically 0 up to rounding

----------------------------

# f = x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1

#L, the resolvent
t^24 + 1474*t^23 + 6114251*t^22 + 10724770749*t^21 + 18040324115451*t^20 + 29907874481435379*t^19 + 86063430612699385021*t^18 + 102144926555868040226129*t^17 + 147500759199880591941664371*t^16 + 78375907726421115644985991554*t^15 + 44662944937519276752622192061756*t^14 + 17322292693580102299584668393879344*t^13 + 3047673626051190489685814168121379131*t^12 + 1217850729197486363604707997579470936119*t^11 + 217282254243426951153060566490668577571956*t^10 + 99002817265467419551817471642071947652251754*t^9 + 55399771384254577925910775957283289857534272021*t^8 - 754041808545820506432728888992930547422769988121*t^7 + 135116488483935929438098470915800345116358257912121*t^6 - 1442394373978671935699518212086988256874323835978321*t^5 + 9545081960457616833405116434345188372164230598497251*t^4 - 30016831938171697345762617941814941836721341292033151*t^3 + 43154954977187854107420144978125462305652904027647751*t^2 - 4713582185628578674191379199196538639650711221001*t + 49325423123140236141788511429040664561845978951

# psis, -1 and the roots of L:
[-1.00000000000000, -398.479673453110 + 47.5914892045037*I, -398.479673453111 - 47.5914892045037*I, -91.0203265468894 + 390.853297485511*I, -91.0203265468895 - 390.853297485511*I]

# purported root r:
1/5*(1/2*(1/2)^(1/5)*(385*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 440*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 + 3520*sqrt(5) + 3520*I*sqrt(2*sqrt(5) + 10) - 7744)^(1/5)) + 1/5*(-55/32*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 165/16*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 - 55*sqrt(5) - 55*I*sqrt(2*sqrt(5) + 10) - 231)^(1/5) + 1/5*(-55/32*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 55/8*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 + 275/4*sqrt(5) + 275/4*I*sqrt(2*sqrt(5) + 10) - 979/4)^(1/5) + 1/5*(-165/64*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 - 385/16*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 - 275/4*sqrt(5) - 275/4*I*sqrt(2*sqrt(5) + 10) - 1529/4)^(1/5) - 1/5

# f(r):  19.7966150028646 + 3.55062195823940e-15*I, which is probably real, but is decidedly _not_ 0.