#
#  Parametric equations of sectic curves of total
#   Milnor number 19 with A_n singularities only
#
#  Author: Stepan Orevkov (orevkov(at)math.ups-tlse.fr)
#  Last update: December 18, 2014
#
#==================================================
# No 1. A19
#
# t = 0,infty --> A19 at (0:0:1) tangent to x=0

alias(a=RootOf(Z^2-6*Z+4));
alias(w=RootOf(Z^2+Z+1));
x := w*t^2 + a*t^3 - w^2*t^4;
y := 2*(t+t^5) - 5*a*(t^2-t^4) + (6+3*a)*t^3;
z :=  (w^2-w*t^6) + 4*a*(w*t+w^2*t^5) + (81*a/2-34)*(t^2-t^4) + (63*a-54)*t^3;

#==================================================
# No 2. A18 + A1
#
# t = infty  --> A18 at (0:0:1) tangent to y=0
# roots of p --> A1  at (0:1:0)

alias(a=RootOf(Z^3-2*Z-2));
p := 3*t^2 - 12*t + 5*a^2 + 8*a + 17;
x := (t + 3 - a)*(t + 3 + a)*p;
y := t^2 + 3*a^2+4*a-1;
z := (9*t^4 + 72*t^3 - (52*a^2+100*a-232)*t^2 - (204*a^2+528*a-420)*t
     - 92*a^2-326*a+587)*p;

#==================================================
# No 3. (A17 + A2)
#
# t = infty  --> A2  at (1:0:0) tangent to y=0
# roots of p --> A17 at (0:0:1) tangent to x=0
# The curve is symmetric under (x:y:z) --> (x:-y:z)

p:=t^2-3;    x:=(t^2+9)*p^2;     y:=t*p;     z:=t^4-12*t^2+3;

#==================================================
# No 4. A16 + A3
#
# t = infty  --> A16 at (0:0:1) tangent to y=0
# roots of p --> A3  at (0:1:0) tangent to z=0

a := (9+sqrt(17))/8;
p := t^2 + t - a+2/3;
x := (t^2 - 3*t - 6+11*a)*p;
y := t^2 - a;
z := (t^2 - 6*t + 6+5*a)*p^2;

#==================================================
# No 5. A16 + A2 + A1
#
# t = infty  --> A16 at (0:1:0)
# t = 0      --> A2  at (0:0:1)
# roots of p --> A1  at (1:0:0)

alias(a=RootOf(Z^3-Z^2+Z-3));
p := 2*t^2 + (8+7*a-7*a^2)*(2*t+1);
x := (t^2 + (8+13*a-13*a^2)*t -(1672-3049*a+1261*a^2)/14)*t^2;
y := (2*t-1+16*a-13*a^2)*(2*t+9+12*a-15*a^2)*t^2*p;
z := (2*t^2 + (a-a^2)*(12*t-9))*p;

#==================================================
# No 6. A15 + A4
#
# t = infty  --> A4  at (0:0:1) tangent to y=0
# roots of p --> A15 at (0:1:0) tangent to z=0

p := 15*t^2 - 7 + 6*I;
x := (145*t^2 - 580*I*t - 133 - 146*I)*p;
y := (1649*t + 724 - 465*I)*(170*t - 7 - 164*I);
z := (5*t^2 + 10*I*t + 1 + 2*I)*p^2;

#==================================================
# No 7. A14 + A4 + A1
#
# t = infty  --> A14 at (0:0:1)
# t = 0      --> A4  at (0:1:0)
# roots of p --> A1  at (1:0:0)

alias(a=RootOf(2*Z^6 - 6*Z^5 + 10*Z^4 - 5*Z^3 - 10*Z^2 + 4*Z + 8));
p:=t^2 - 12*t - (338*a^5-638*a^4+458*a^3+771*a^2-4382*a-2592)/187;
x:=(t^2 + (4*a-12)*t - 2*a^5+6*a^4-2*a^3-3*a^2-14*a+24)*t^2;
y:=(t^2 + 4*a*t -(10*a^5-22*a^4+50*a^3-73*a^2-38*a+16)/11)*p;
z:=(t^2+8*(a-1)*t-(10*a^5-22*a^4+50*a^3-73*a^2+50*a+16)/11)*p*t^2;

#==================================================
# No 8. (A14 + A2) + A3
#
# t = infty  --> A14 at (0:0:1) tangent to x=0
# t = 2      --> A2  at (1:3:-1)
# roots of p --> A3  at (1:0:0) tangent to z=0

p:=t^2+2*t-9;   x:=t^2-5;   y:=(t^2+2*t-5)*p;   z:=(t^2+4*t-11)*p^2;

#==================================================
# No 9. (A14 + A2) + A2 + A1
#
# t = infty  --> A14 at (0:0:1)
# t = 0      --> A2  at (1:0:0)
# t = 1      --> A2  at (1:-7:1)
# roots of p --> A1  at (0:1:0)

p:=t^2+5*t-5;   x:=(2*t^2-3)*p;   y:=(t^2+5*t+1)*t^2;   z:=(t^2+4*t-6)*p*t^2;

#==================================================
# No 10. A13 + A6
#
# t = infty  --> A6  at (0:0:1) tangent to y=0
# roots of p --> A13 at (0:1:0) tangent to z=0

alias(a=RootOf(Z^4+7));
p := 3*t^2 + 7*a^3-5*a^2-13*a+39;
x := (3*t - 2*a^3+2*a^2-6*a+2)*(3*t + 2*a^3-a^2-2*a+3)*p;
y := 3*t^2+4*(a^2-8*a+5)*t+(-1037*a^3+2143*a^2-1105*a-2757)/33;
z := ( 3*t^2 - (a^2-8*a+5)*t + (31*a^3+15*a^2-149*a+235)/3 )*p^2;

#==================================================
# No 11. A13 + A4 + A2
#
# t = infty  --> A4  at (0:0:1)
# root  if q --> A2  at (1:0:0)
# roots of p --> A1  at (0:1:0)

alias(a=RootOf(Z^2-21));
p := 2*t^2+11*a-39;
q := 2*t+3*a-6;
x := (t+a-3)*(t-3-2*a)*p;
y := (2*t-5*a-9)*(2*t-3*a+9)*q^2;
z := (t-a+3)*(t+3)*p*q^2;

#==================================================
# No 12. A12 + A7
#
# t = infty  --> A12 at (0:1:0) tangent to z=0
# roots of p --> A7  at (0:0:1) tangent to y=0

p := 3*t^2 + 1;
x := (13*t^2 + (-8+12*I)*t + 7-4*I)*p;
y := (13*t^2 + (-20+30*I)*t -11-16*I)*p^2;
z := (13*t + 3+2*I)*(13*t + 1-8*I);

#==================================================
# No 13. A12 + A6 + A1
#
# t = infty  --> A12 at (0:0:1)
# t = 0      --> A6  at (0:1:0)
# roots of p --> A1  at (1:0:0)

alias(a=RootOf(7*Z^3-16*Z^2+12*Z-4));
p := 3*t^2 + (12*a-18)*t + 28*a^2 - 53*a + 36;
x := (t^2+(4*a-2)*t+14*a^2-15*a+4)*t^2;
y := (t^2 + 4*t - 7*a + 4)*p;
z := (t^2 + (4*a+4)*t + 13*a)*p*t^2;

#==================================================
# No 14. A12 + A4 + A3
#
# t = infty  --> A12 at (0:1:0) tangent to z=0
# t = 0      --> A4  at (0:0:1) tangent to y=0
# roots of p --> A3  for  p = 85*t^2 - 442*t + 507

x:=(5*t^2-12*t-13)*t^2;  y:=(25*t^2-20*t-221)*t^4;  z:=t^2-4*t+3;

#==================================================
# No 15. A12 + A4 + A2 + A1
#
# t = infty  --> A12 at (0:0:1) tangent to y=0
# t = 0      --> A4  at (0:1:0) tangent to z=0
# t = 30*a^2-159/2*a+55/2  -->  A2
# roots of p --> A1
#   for p = t^2 + (60*a^2-141*a+64)*t + 3828/5*a^2-1851*a+772

alias(a=RootOf(15*Z^3-48*Z^2+40*Z-10));
x := (t^2 + 9*t -24*a^2+24*a+5)*t^2;
y := t^2 + 9*a*t + 84*a^2-165*a+50;
z := (t^2 + (-9*a+18)*t -51*a^2+24*a+41)*t^4;

#==================================================
# No 16. A11 + 2A4
#
# roots of p --> A11 at (0:1:0) tangent to z=0
# roots of q --> 2A4 on the line y=0

a:=sqrt(2); i:=I;
p := t^2+(2*a*i+a-3*i-2)*t+a+i-1; q := 2*t^2-(5*i+3)*t-2;
x := (17*t^4 +(34*a*i-9*a-34*i-20)*t^3+(136*a*i+49*a+34*i-148)*t^2
     +(-42*a*i+127*a+122*i+48)*t -37*a*i-13*a-18*i+24)*p;
y := (4*a+34*i-27)*(t+2*a*i+a-3*i-1)*(17*t-6*a*i-7*a-15*i-9)*q^2;
z := (2*t-i)*p^2;

#---------------------------------------------------
# Another parametrization of the same (sic!) curve:
#
# 0,infty    --> A11 at (0:1:0) tangent to z=0
# roots of q --> 2A4 on the line y=0

alias(b=RootOf(b^4 + 14*b^2 + 17));
q := 16*t^2 - (b^3-7*b^2+7*b-17)*t + b^3+b^2+15*b-1;
x := (32*t^4-(7*b^3-49*b^2+17*b-135)*t^3-(28*b^3-44*b^2+52*b-4)*t^2
     +(10*b^3+2*b^2-26*b-50)*t + 2*b^3+4*b^2+18*b+32)*t;
y :=(b^2-2*b+9)*(32*t^2+(28*b^3+8*b^2+28*b+16)*t+2*b^3-b^2+24*b+19)*q^2;
z := (b^2+2*b+9)*(2*t^2-(b^2-1)*t-2)*t^2;

#==================================================
# No 17. (A11 + 2A2) + A4
#
# t = infty  --> A4  at (1:0:0) tangent to z=0
# roots of p --> A11 at (0:0:1) tangent to x=0
# roots of q --> 2A2 for q = 3*t^2 + 2

p:=3*t^2+9*t+8;  x:=(3*t^2-10)*p^2;  y:=(3*t^2+9*t+5)*p;  z:=3*t^2+3*t+2;

#==================================================
# No 18. A10 + A9
#
# t = infty  --> A10 at (0:1:0) tangent to z=0
# roots of p --> A9  at (0:0:1) tangent to y=0

alias(a=RootOf(Z^2-5)); p := t^2 - 11 - 22/3*a;
x := (t - 3 - 2*a)*(t - 5 + 4*a)*p;
y := (t^2 - (32-8*a)*t + 149 + 70*a)*p^2;
z :=  t^2 + (16-4*a)*t + 161 - 226/5*a;

#==================================================
# No 19. A10 + A8 + A1
#
# t = infty  --> A10 at (0:0:1)
# t = 0      --> A8  at (0:1:0)
# roots of p --> A1  at (1:0:0)

alias(a=RootOf(Z^3-4*Z^2+8*Z-4));
p := t^2 + (8*a-6)*t + 9*a^2-29*a+16;
x := ( t^2 + (4*a-2)*t + 15*a^2-45*a+24 )*t^2;
y := ( t^2 - (4*a-4)*t + 5*a^2-9*a+4 )*p;
z := ( t^2 + 4*t - 3*a^2 + 3*a )*p*t^2;

#==================================================
# No 20. A10 + A7 + A2
#
# t = infty       --> A10 at (0:1:0) tangent to z=0
# roots of p      --> A7  at (0:0:1) tangent to y=0
# t = (3+27*a)/22 --> A2

alias(a=RootOf(Z^2-3));  p := 2*t^2 - 6 + a;
x := (22*t^2 - (12 + 20*a)*t + 30 + 39*a)*p;
y := (22*t^2 - (42 + 26*a)*t - 6 + 45*a)*p^2;
z :=  22*t^2 + (18 - 14*a)*t - (300 + 159*a)/11;

#==================================================
# No 21. A10 + A6 + A3
#
# t = infty  --> A10 at (0:0:1) tangent to y=0
# t = 0      --> A6  at (0:1:0) tangent to z=0
# roots of p --> A3  for  p = 22*t^2+(27-3*a)*t+9-3*a

alias(a=RootOf(Z^2+7));
x := (4*t+1-a)*(t+1)*t^2;
y := 22*t^2+(56-16*a)*t-3-7*a;
z := (22*t^2+(-1+5*a)*t-5+3*a)*t^4;

#==================================================
# No 22. A10 + A6 + A2 + A1
#
# t = infty   --> A10 at (0:0:1) tangent to y=0
# t = 0       --> A6  at (0:1:0) tangent to z=0
# t = 4*a^2-3 --> A2
# roots of p  --> A1  for  p = t^2 + (2*a^2+6*a+3)*t - 26*a^2+78*a+123

alias(a=RootOf(Z^3-Z^2+3));
x := ( t^2 + 9*t - 22*a^2+30*a+21 )*t^2;
y :=   t^2 - (6*a^2-6*a+9)*t + 50*a^2 - 78*a + 75;
z := ( t^2 + (6*a^2-6*a+27)*t -22*a^2+66*a-33 )*t^4;

#==================================================
# No 23. A10 + A5 + A4
#
# t = infty  --> A10 at (0:1:0)
# t = 0      --> A4  at (1:0:0)
# roots of p --> A1  at (0:0:1)

alias(a=RootOf(Z^2-15));
p := 5*t^2 + (3*a+5)*t + (70-24*a)/17;
x := (t+a-4)*(t+1)*p;
y := (t^2 + (4*a-12)*t - 2*a+8)*p*t^2;
z := (5*t^2 + (8*a-10)*t + (110-2*a)/7 )*t^2;

#==================================================
# No 24. A10 + 2A4 + A1
#
# t = infty  --> A10 at (0:1:0) tangent to z=0
# roots of q --> 2A4 on the line y=0
# roots of p --> A1  on the line x=0

alias(a=RootOf(Z^3-Z^2-Z-1));
p := t^2 + (-12*a^2+18*a+8)*t - 39*a^2+12*a+116;
q := t^2-5*a^2+20;
x := (t - a)*(t + 4*a^2-5*a-4)*p;
y := (t^2 + (-20*a^2+30*a+8)*t +45*a^2+20*a-140)*q^2;
z := t^2 + (4*a^2-6*a)*t + a^2+4*a+4;

#==================================================
# No 25. A10 + A4 + A3 + A2
#
# t = infty  --> A10 at (0:1:0) tangent to z=0
# t = 0      --> A4  at (0:0:1) tangent to y=0
# t = 4      --> A2  at (32:256:1)
# roots of p --> A1  for p = 20*t^2-55*t-121

x:=(7*t^2-35*t+22)*t^2;  y:=(8*t^2-52*t+77)*t^4;  z:=2*t^2-7*t-7;

#==================================================
# No 26. A10 + A4 + 2A2 + A1
#
# t = infty  --> A10 at (0:0:1) tangent to y=0
# t = 0      --> A4  at (0:1:0) tangent to z=0
# roots of q --> 2A2 for q = t^2 + 3*t +*(15-a)/6
# roots of p --> A1  for p = t^2+(a-4)*t+(15-23*a)/6

alias(a=RootOf(Z^2-5));
x := (6*t^2 + 6*t + 9*a-25)*t^2;
y :=  6*t^2 + 6*a*t + 5*(a-1);
z := (6*t^2 + (12-6*a)*t - 11*(a-1))*t^4;

#==================================================
# No 27. A9 + A6 + A4
#
# t = infty  --> A4  at (0:1:0)
# t = 0      --> A6  at (0:0:1)
# roots of p --> A9  at (1:0:0)

alias(a=RootOf(Z^3-5*Z-5));
p := 3*t^2 - (4*a^2+4*a-7)*t + 4*a^2 + 13*a + 20;
x := ( 3*t^2 - (4*a^2+4*a-25)*t - (274*a^2-365*a-1510)/31 )*t^2;
y := (t - a^2 - a + 1)*(3*t - a^2 - a - 5)*t^2*p;
z := (t^2 + 6*t - 2*a^2 - 11*a - 10)*p;

#==================================================
# No 28. A9 + 2A4 + A2
#
# t = infty  --> A2  at (0:0:1) tangent to x=0
# roots of q --> 2A4 on the line 200*y=7*z where q = 2*t^2+15
# roots of p --> A9  at (0:1:0) tangent to z=0

p:=2*t^2-5;   x:=p*t;   y:=4*t^4-80*t^2+15;   z:=(2*t^2+75)*p^2;

#==================================================
# No 29. (2A8) + A3
#
# t = infty  --> A8  at (0:0:1) tangent to x=0
# t = 0      --> A8  at (1:0:0) tangent to z=0
# roots of p --> A3  at (1:1:-1) tangent to z+x=0 where p=t^2+2*t-1

x:=3*t^2-4*t+1;    y:=(3*t^2+4*t-3)*t^2;    z:=(t^2+4*t+3)*t^4;

#==================================================
# No 30. A8 + A7 + A4
#
# t = infty   --> A8  at (0:1:0) tangent to z=0
# roots of p  --> A7  at (0:0:1) tangent to y=0
# t = (I-1)/2 --> A4

p := 34*t^2 - 8+19*I;
x := (30*t^2+20*(1-I)*t-4-3*I)*p;
y := (10*t^2-(2+6*I)*t+5*I)*p^2;
z := (10*t+9-3*I)*(30*t+19-13*I);

#==================================================
# No 31. A8 + A6 + A4 + A1
#
# t = infty  --> A8  at (0:0:1)
# t = 0      --> A6  at (0:1:0)
# t = 0      --> A4  at (1:0:0)
# roots of p --> A1  for p = t^2+(a^2-4*a+5)*t + (43*a^2-104*a+120)/7

alias(a=RootOf(Z^3 - Z^2 - Z + 5));
x := (15*t^2 + (5-20*a+5*a^2)*t - 70 + 64*a - 23*a^2)*t^2;
y := (21*t^2 + (49-28*a+7*a^2)*t -18 + 24*a - 11*a^2)*(t-1)^2;
z := (5*t^2 + (1-4*a-a^2)*t - 10+5*a^2)*t^2*(t-1)^2;

#==================================================
# No 32. (A8 + A5 + A2) + A4
#
# t = infty   --> A8  at (1:0:0) tangent to y-z=0
# t = 0       --> A4  at (0:1:0) tangent to (3+I)*x+4*z=0
# roots of p  --> A7  at (0:0:1) tangent to 5*x+8*y=0
# t = 1       --> A2  at (-1:1:(7+4*I)/9)

p := t^2+2*I*t-(8+4*I)/5;
x := (t^2 - 4)*p*t^2;
y := (3*t^2 - 2*t + 2 )*p;
z := (3*t^2 - (2-6*I)*t - 4-4*I)*t^2;

#==================================================
# No 33. (A8 + 3A2) + A4 + A1
#
# t = infty  --> A8  at (0:1:0) tangent to x=0
# t = 0      --> A4  at (0:0:1) tangent to y=0
# roots of q --> 3A2 where q=t^3-3t-3
# roots of p --> A1  at (27:486:2) where p=t^2+3*t-9

x:=(t^2+t-3)*t^2;    y:=(t^2+3*t+3)*t^4;    z:=t^2-t-1;

#==================================================
# No 34. A7 + 2A6
#
# 0,infty    --> A7  at (0:1:0) tangent to z=0
# roots of q --> 2A6 on the line y=0

alias(a=RootOf(Z^2+7)); alias(b=RootOf(Z^2+3));
q:=8*t^2 -(a*b+3*a+5*b-9)*t-a*b-3*a-5*b+5;
x:=4*(a+1)*(16*t^4-(a*b+a+b-23)*t^3-(12*a*b-20*a-28*b-60)*t^2
          -(23*a*b+2*a+3*b-106)*t-10*a*b+46)*t;
y:=(a*b+5)*(8*t^2 +(2*a*b-16*a-26*b-12)*t +5*a*b+15*a+23*b-23)*q^2;
z:=2*(a+1)^2*(a*b-5)*(2*t^2-(a-3)*t+1)*t^2;

h := 4*X^2+4*a*X-7+7*a;
f := 1/4*Y^4 + (X^2+(2-6*a)*X+58+(23/2)*a)*Y^3
 +(X^4 -(10+26*a)*X^3 + (162-99*a)*X^2 + (1883-265*a)*X + 1799+1428*a)*Y^2
 -(a+1)*(7*X^3 + (79-2*a)*X^2 + (182+9*a)*X + 658 + 336*a)*h*Y
 -((15+7*a)*X^2 + (175+43*a)*X +623-21*a)*h^2;

# Check if the implicit equation defines the same curve:

F:=factor(subs(X=X/Z,Y=Y/Z,f)*Z^6); # homogeneization of f
factor(subs(X=x,Y=y,Z=z,F));        # should be equal to 0

#==================================================
# No 35. A7 + A6 + A4 + A2
#
# roots of p --> A7  at (0:1:0)
# infty      --> A6  at (1:0:0)
# 0          --> A4  at (0:0:1)
# 4          --> A2

alias(a=RootOf(Z^2-21));
p := 2*t^2-(9-a)*t+(2*a-78)/5;
x := (2*t^2-(23-3*a)*t+116-24*a)*t^2*p;
y := (t^2-5*t-2+4/3*a)*t^2;
z := (2*t^2-(1+a)*t+4-8/3*a)*p;

#==================================================
# No 36. A7 + 2A4 + 2A2
#
# 0,infty    --> A7  at (0:1:0) tangent to z=0
# roots of q --> 2A4 on the line y=0
# a, a/2     --> 2A2 on the line 4*x-y=0

alias(a=RootOf(Z^2+Z+1));
q := 2*t^2 + (2+a)*t - a^2;
x := a*t*(2*t^2 - 2*a*t + a^2)*(4*t^2 + 5*t + 2);
y := (2*t^2 - t + 1)*q^2;
z := a^2*t^2*(2*t^2 + 2*t + 1);

h := X^2+11*X-1;
f := 32*Y^4 + (-16*X^2 - 288*X + 248)*Y^3
 + (2*X^4 + 96*X^3 +570*X^2 - 2816*X - 198)*Y^2
 + (-6*X^3 + 16*X^2 + 948*X + 238)*h*Y - (9*X^2 + 105*X + 49)*h^2;

# Check if the implicit equation defines the same curve:

F:=factor(subs(X=X/Z,Y=Y/Z,f)*Z^6); # homogeneization of f
factor(subs(X=x,Y=y,Z=z,F));        # should be equal to 0


#==================================================
# No 37. 3A6 + A1
#
# t = infty   --> A6  at (0:0:1) tangent to x-3*y=0
# t = 0       --> A6  at (0:1:0) tangent to y-3*z=0
# t = 1       --> A6  at (1:0:0) tangent to z-3*x=0
# roots of p  --> A1  at (1:1:1) for p=t^2-t+1

x := (3*t^2-3*t+1)*t^2;
y := (t^2+t+1)*(t-1)^2;
z := (t^2-3*t+3)*t^2*(t-1)^2;

#==================================================
# No 38. 2A6 + A4 + A2 + A1
#
# infty      --> A4 at (0:0:1) tangent to y=0
# 0          --> A2 at (0 : 1 : 26-6a)
# roots of q --> 2A6 on the line z=0
# roots of p --> A1 where p=2*t^2+(3-a)*t+6-4/3*a

alias(a=RootOf(Z^2-21));
q := 2*t^2 + 6*t + 1-a;
x := ( 2*t^2 + (17+a)*t - 2 - 4*a )*t^2;
y :=  10*t^2 + (19-a)*t + 10;
z := (2*t^2 - (1+a)*t + 8-2*a)*q^2;

#==================================================
# No 39. A6 + A5 + 2A4
#
# infty      --> A6 at (0:1:1)
# roots of p --> A5 at (0:0:1) tangent to y=0
# roots of q --> 2A4 on the line z=0

alias(a=RootOf(Z^2-7));
p := t^2-21-8*a;         q := t^2+(6+a)*t+7+3*a;
x := (t+2+a)*(t+4+a)*p;
y := (t^2+12*t+15+8*a)*p^2;
z := (t^2-2*a*t-(7+2*a)/3)*q^2;