433.sagews

#9.2.1
var('d')
var('l')
var('b')
var('r')
var('w')
J=[[-d-l,d,0],[r,-1-l,sqrt(b*(r-1))],[sqrt(b*(r-1)),0,-b-l]]
MJ=Matrix(J)
CE=MJ.determinant()
MJ

[ -l - d d 0] [ r -l - 1 sqrt(b*(r - 1))] [sqrt(b*(r - 1)) 0 -l - b]

-1*expand(CE)

-d*l*r - 2*b*d*r + l^3 + d*l^2 + b*l^2 + l^2 + b*d*l + d*l + b*l + 2*b*d

solve([CE==0],l)

[l == (-sqrt(3)*I/2 - 1/2)*(sqrt(-4*d^3*r^3 - (d^4 + (26*b - 10)*d^3 + (-71*b^2 + 26*b + 1)*d^2)*r^2 + (-(6*b - 2)*d^4 - (-4*b^2 - 34*b + 8)*d^3 - (-14*b^3 + 134*b^2 - 34*b - 2)*d^2 - (8*b^4 - 14*b^3 - 4*b^2 + 6*b)*d)*r + (-b^2 + 6*b - 1)*d^4 + (2*b^3 - 8*b^2 - 8*b + 2)*d^3 + (-b^4 - 8*b^3 + 57*b^2 - 8*b - 1)*d^2 + (6*b^4 - 8*b^3 - 8*b^2 + 6*b)*d - b^4 + 2*b^3 - b^2)/(6*sqrt(3)) - (d^2*(9*r - 3) + d*(9*r - 3) + b*(d*(39 - 45*r) - 3*d^2 - 3) + 2*d^3 + b^2*(-3*d - 3) + 2*b^3 + 2)/54)^(1/3) + (sqrt(3)*I/2 - 1/2)*(d*(3*r - 1) + d^2 + b*(-d - 1) + b^2 + 1)/(9*(sqrt(-4*d^3*r^3 - (d^4 + (26*b - 10)*d^3 + (-71*b^2 + 26*b + 1)*d^2)*r^2 + (-(6*b - 2)*d^4 - (-4*b^2 - 34*b + 8)*d^3 - (-14*b^3 + 134*b^2 - 34*b - 2)*d^2 - (8*b^4 - 14*b^3 - 4*b^2 + 6*b)*d)*r + (-b^2 + 6*b - 1)*d^4 + (2*b^3 - 8*b^2 - 8*b + 2)*d^3 + (-b^4 - 8*b^3 + 57*b^2 - 8*b - 1)*d^2 + (6*b^4 - 8*b^3 - 8*b^2 + 6*b)*d - b^4 + 2*b^3 - b^2)/(6*sqrt(3)) - (d^2*(9*r - 3) + d*(9*r - 3) + b*(d*(39 - 45*r) - 3*d^2 - 3) + 2*d^3 + b^2*(-3*d - 3) + 2*b^3 + 2)/54)^(1/3)) + (-d - b - 1)/3, l == (sqrt(3)*I/2 - 1/2)*(sqrt(-4*d^3*r^3 - (d^4 + (26*b - 10)*d^3 + (-71*b^2 + 26*b + 1)*d^2)*r^2 + (-(6*b - 2)*d^4 - (-4*b^2 - 34*b + 8)*d^3 - (-14*b^3 + 134*b^2 - 34*b - 2)*d^2 - (8*b^4 - 14*b^3 - 4*b^2 + 6*b)*d)*r + (-b^2 + 6*b - 1)*d^4 + (2*b^3 - 8*b^2 - 8*b + 2)*d^3 + (-b^4 - 8*b^3 + 57*b^2 - 8*b - 1)*d^2 + (6*b^4 - 8*b^3 - 8*b^2 + 6*b)*d - b^4 + 2*b^3 - b^2)/(6*sqrt(3)) - (d^2*(9*r - 3) + d*(9*r - 3) + b*(d*(39 - 45*r) - 3*d^2 - 3) + 2*d^3 + b^2*(-3*d - 3) + 2*b^3 + 2)/54)^(1/3) + (-sqrt(3)*I/2 - 1/2)*(d*(3*r - 1) + d^2 + b*(-d - 1) + b^2 + 1)/(9*(sqrt(-4*d^3*r^3 - (d^4 + (26*b - 10)*d^3 + (-71*b^2 + 26*b + 1)*d^2)*r^2 + (-(6*b - 2)*d^4 - (-4*b^2 - 34*b + 8)*d^3 - (-14*b^3 + 134*b^2 - 34*b - 2)*d^2 - (8*b^4 - 14*b^3 - 4*b^2 + 6*b)*d)*r + (-b^2 + 6*b - 1)*d^4 + (2*b^3 - 8*b^2 - 8*b + 2)*d^3 + (-b^4 - 8*b^3 + 57*b^2 - 8*b - 1)*d^2 + (6*b^4 - 8*b^3 - 8*b^2 + 6*b)*d - b^4 + 2*b^3 - b^2)/(6*sqrt(3)) - (d^2*(9*r - 3) + d*(9*r - 3) + b*(d*(39 - 45*r) - 3*d^2 - 3) + 2*d^3 + b^2*(-3*d - 3) + 2*b^3 + 2)/54)^(1/3)) + (-d - b - 1)/3, l == (sqrt(-4*d^3*r^3 - (d^4 + (26*b - 10)*d^3 + (-71*b^2 + 26*b + 1)*d^2)*r^2 + (-(6*b - 2)*d^4 - (-4*b^2 - 34*b + 8)*d^3 - (-14*b^3 + 134*b^2 - 34*b - 2)*d^2 - (8*b^4 - 14*b^3 - 4*b^2 + 6*b)*d)*r + (-b^2 + 6*b - 1)*d^4 + (2*b^3 - 8*b^2 - 8*b + 2)*d^3 + (-b^4 - 8*b^3 + 57*b^2 - 8*b - 1)*d^2 + (6*b^4 - 8*b^3 - 8*b^2 + 6*b)*d - b^4 + 2*b^3 - b^2)/(6*sqrt(3)) - (d^2*(9*r - 3) + d*(9*r - 3) + b*(d*(39 - 45*r) - 3*d^2 - 3) + 2*d^3 + b^2*(-3*d - 3) + 2*b^3 + 2)/54)^(1/3) + (d*(3*r - 1) + d^2 + b*(-d - 1) + b^2 + 1)/(9*(sqrt(-4*d^3*r^3 - (d^4 + (26*b - 10)*d^3 + (-71*b^2 + 26*b + 1)*d^2)*r^2 + (-(6*b - 2)*d^4 - (-4*b^2 - 34*b + 8)*d^3 - (-14*b^3 + 134*b^2 - 34*b - 2)*d^2 - (8*b^4 - 14*b^3 - 4*b^2 + 6*b)*d)*r + (-b^2 + 6*b - 1)*d^4 + (2*b^3 - 8*b^2 - 8*b + 2)*d^3 + (-b^4 - 8*b^3 + 57*b^2 - 8*b - 1)*d^2 + (6*b^4 - 8*b^3 - 8*b^2 + 6*b)*d - b^4 + 2*b^3 - b^2)/(6*sqrt(3)) - (d^2*(9*r - 3) + d*(9*r - 3) + b*(d*(39 - 45*r) - 3*d^2 - 3) + 2*d^3 + b^2*(-3*d - 3) + 2*b^3 + 2)/54)^(1/3)) + (-d - b - 1)/3]

CE=CE.substitute(l=w*I)
CE=CE.substitute(r=(d+b+3)/(d-b-1))
CE

(-1*I*w - 1)*(-1*I*w - b)*(-1*I*w - d) - d*((d + b + 3)*(-1*I*w - b)/(d - b - 1) - b*((d + b + 3)/(d - b - 1) - 1))

assume(d>b+1)
assume(w,'real')

solve([CE==0],w)

[w == (-sqrt(3)*I/2 - 1/2)*(sqrt(b^2*d^7 + (-5*b^3 + 13*b^2 + 24*b)*d^6 + (10*b^4 - 38*b^3 - 51*b^2 + 40*b + 16)*d^5 + (-10*b^5 + 14*b^4 + 61*b^3 + 281*b^2 + 368*b + 16)*d^4 + (5*b^6 + 48*b^5 - 188*b^4 - 1340*b^3 - 1385*b^2 + 288*b + 240)*d^3 + (-b^7 - 51*b^6 + 116*b^5 + 1252*b^4 + 1889*b^3 + 395*b^2 - 424*b - 16)*d^2 + (14*b^7 + 39*b^6 - 32*b^5 - 130*b^4 - 54*b^3 + 43*b^2 + 24*b)*d - b^7 - b^6 + 2*b^5 + 2*b^4 - b^3 - b^2)/(6*sqrt(3)*(d - b - 1)^(3/2)) - I*(-2*d^4 - 4*d^3 + b*(-4*d^3 + 12*d + 8) - b*(-9*d^3 + 9*d^2 - 153*d + 9) - 36*d^2 + b^2*(12*d + 12) - b^3*(9*d + 9) + b^3*(4*d + 8) - 32*d - b^2*(18 - 72*d) + 2*b^4 + 2)/(-54*d + 54*b + 54))^(1/3) + (sqrt(3)*I/2 - 1/2)*(-d^3 - d^2 + b*(-d^2 + 2*d + 3) - b*(-3*d^2 + 6*d + 3) - b^2*(3*d + 3) + b^2*(d + 3) - 11*d + b^3 + 1)/((9*d - 9*b - 9)*(sqrt(b^2*d^7 + (-5*b^3 + 13*b^2 + 24*b)*d^6 + (10*b^4 - 38*b^3 - 51*b^2 + 40*b + 16)*d^5 + (-10*b^5 + 14*b^4 + 61*b^3 + 281*b^2 + 368*b + 16)*d^4 + (5*b^6 + 48*b^5 - 188*b^4 - 1340*b^3 - 1385*b^2 + 288*b + 240)*d^3 + (-b^7 - 51*b^6 + 116*b^5 + 1252*b^4 + 1889*b^3 + 395*b^2 - 424*b - 16)*d^2 + (14*b^7 + 39*b^6 - 32*b^5 - 130*b^4 - 54*b^3 + 43*b^2 + 24*b)*d - b^7 - b^6 + 2*b^5 + 2*b^4 - b^3 - b^2)/(6*sqrt(3)*(d - b - 1)^(3/2)) - I*(-2*d^4 - 4*d^3 + b*(-4*d^3 + 12*d + 8) - b*(-9*d^3 + 9*d^2 - 153*d + 9) - 36*d^2 + b^2*(12*d + 12) - b^3*(9*d + 9) + b^3*(4*d + 8) - 32*d - b^2*(18 - 72*d) + 2*b^4 + 2)/(-54*d + 54*b + 54))^(1/3)) + I*(d + b + 1)/3, w == (sqrt(3)*I/2 - 1/2)*(sqrt(b^2*d^7 + (-5*b^3 + 13*b^2 + 24*b)*d^6 + (10*b^4 - 38*b^3 - 51*b^2 + 40*b + 16)*d^5 + (-10*b^5 + 14*b^4 + 61*b^3 + 281*b^2 + 368*b + 16)*d^4 + (5*b^6 + 48*b^5 - 188*b^4 - 1340*b^3 - 1385*b^2 + 288*b + 240)*d^3 + (-b^7 - 51*b^6 + 116*b^5 + 1252*b^4 + 1889*b^3 + 395*b^2 - 424*b - 16)*d^2 + (14*b^7 + 39*b^6 - 32*b^5 - 130*b^4 - 54*b^3 + 43*b^2 + 24*b)*d - b^7 - b^6 + 2*b^5 + 2*b^4 - b^3 - b^2)/(6*sqrt(3)*(d - b - 1)^(3/2)) - I*(-2*d^4 - 4*d^3 + b*(-4*d^3 + 12*d + 8) - b*(-9*d^3 + 9*d^2 - 153*d + 9) - 36*d^2 + b^2*(12*d + 12) - b^3*(9*d + 9) + b^3*(4*d + 8) - 32*d - b^2*(18 - 72*d) + 2*b^4 + 2)/(-54*d + 54*b + 54))^(1/3) + (-sqrt(3)*I/2 - 1/2)*(-d^3 - d^2 + b*(-d^2 + 2*d + 3) - b*(-3*d^2 + 6*d + 3) - b^2*(3*d + 3) + b^2*(d + 3) - 11*d + b^3 + 1)/((9*d - 9*b - 9)*(sqrt(b^2*d^7 + (-5*b^3 + 13*b^2 + 24*b)*d^6 + (10*b^4 - 38*b^3 - 51*b^2 + 40*b + 16)*d^5 + (-10*b^5 + 14*b^4 + 61*b^3 + 281*b^2 + 368*b + 16)*d^4 + (5*b^6 + 48*b^5 - 188*b^4 - 1340*b^3 - 1385*b^2 + 288*b + 240)*d^3 + (-b^7 - 51*b^6 + 116*b^5 + 1252*b^4 + 1889*b^3 + 395*b^2 - 424*b - 16)*d^2 + (14*b^7 + 39*b^6 - 32*b^5 - 130*b^4 - 54*b^3 + 43*b^2 + 24*b)*d - b^7 - b^6 + 2*b^5 + 2*b^4 - b^3 - b^2)/(6*sqrt(3)*(d - b - 1)^(3/2)) - I*(-2*d^4 - 4*d^3 + b*(-4*d^3 + 12*d + 8) - b*(-9*d^3 + 9*d^2 - 153*d + 9) - 36*d^2 + b^2*(12*d + 12) - b^3*(9*d + 9) + b^3*(4*d + 8) - 32*d - b^2*(18 - 72*d) + 2*b^4 + 2)/(-54*d + 54*b + 54))^(1/3)) + I*(d + b + 1)/3, w == (sqrt(b^2*d^7 + (-5*b^3 + 13*b^2 + 24*b)*d^6 + (10*b^4 - 38*b^3 - 51*b^2 + 40*b + 16)*d^5 + (-10*b^5 + 14*b^4 + 61*b^3 + 281*b^2 + 368*b + 16)*d^4 + (5*b^6 + 48*b^5 - 188*b^4 - 1340*b^3 - 1385*b^2 + 288*b + 240)*d^3 + (-b^7 - 51*b^6 + 116*b^5 + 1252*b^4 + 1889*b^3 + 395*b^2 - 424*b - 16)*d^2 + (14*b^7 + 39*b^6 - 32*b^5 - 130*b^4 - 54*b^3 + 43*b^2 + 24*b)*d - b^7 - b^6 + 2*b^5 + 2*b^4 - b^3 - b^2)/(6*sqrt(3)*(d - b - 1)^(3/2)) - I*(-2*d^4 - 4*d^3 + b*(-4*d^3 + 12*d + 8) - b*(-9*d^3 + 9*d^2 - 153*d + 9) - 36*d^2 + b^2*(12*d + 12) - b^3*(9*d + 9) + b^3*(4*d + 8) - 32*d - b^2*(18 - 72*d) + 2*b^4 + 2)/(-54*d + 54*b + 54))^(1/3) + (-d^3 - d^2 + b*(-d^2 + 2*d + 3) - b*(-3*d^2 + 6*d + 3) - b^2*(3*d + 3) + b^2*(d + 3) - 11*d + b^3 + 1)/((9*d - 9*b - 9)*(sqrt(b^2*d^7 + (-5*b^3 + 13*b^2 + 24*b)*d^6 + (10*b^4 - 38*b^3 - 51*b^2 + 40*b + 16)*d^5 + (-10*b^5 + 14*b^4 + 61*b^3 + 281*b^2 + 368*b + 16)*d^4 + (5*b^6 + 48*b^5 - 188*b^4 - 1340*b^3 - 1385*b^2 + 288*b + 240)*d^3 + (-b^7 - 51*b^6 + 116*b^5 + 1252*b^4 + 1889*b^3 + 395*b^2 - 424*b - 16)*d^2 + (14*b^7 + 39*b^6 - 32*b^5 - 130*b^4 - 54*b^3 + 43*b^2 + 24*b)*d - b^7 - b^6 + 2*b^5 + 2*b^4 - b^3 - b^2)/(6*sqrt(3)*(d - b - 1)^(3/2)) - I*(-2*d^4 - 4*d^3 + b*(-4*d^3 + 12*d + 8) - b*(-9*d^3 + 9*d^2 - 153*d + 9) - 36*d^2 + b^2*(12*d + 12) - b^3*(9*d + 9) + b^3*(4*d + 8) - 32*d - b^2*(18 - 72*d) + 2*b^4 + 2)/(-54*d + 54*b + 54))^(1/3)) + I*(d + b + 1)/3]