実行例（５）
入力は，主に http://www.cs.bath.ac.uk/~djw42/triangular/examplebank.pdf の Examples from [CMXY09]（https://arxiv.org/pdf/0903.5221.pdf）から選びました．
環境は，Core i5-3210M 2.50GHz/8GB/Ubuntu 14.04/Maxima 5.40.0/SBCL 1.3.20 です．
コントローラー qvpeds の第 4 引数が h1 は Hong タイプ，l は Lazard タイプの処理を表し，%pp はそれぞれの projection set です．
(%i4) [(qvpeds([ex],[a,b,c,x],0,h1,r11,0),
qe(bfpcad(ext('(a*x^2+b*x+c = 0))))),%pp]
Evaluation took 0.2600 seconds (0.2620 elapsed) using 30.789 MB.
(%o4) [[[a < root(a,1),true,root(4*a*c-b^2,1) <= c,true],
[a = root(a,1),b < root(b,1),true,true],
[a = root(a,1),b = root(b,1),c = root(c,1),true],
[a = root(a,1),root(b,1) < b,true,true],
[root(a,1) < a,true,c <= root(4*a*c-b^2,1),true]],
[[a],[b],[c,4*a*c-b^2],[a*x^2+b*x+c]]]
(%i5) [(qvpeds([ex],[a,b,c,x],0,l,r11,0),qe(bfpcad(ext('(a*x^2+b*x+c = 0))))),%pp]
Evaluation took 0.2110 seconds (0.2090 elapsed) using 30.792 MB.
(%o5) [[[a < root(a,1),true,root(4*a*c-b^2,1) <= c,true],
[a = root(a,1),b < root(b,1),true,true],
[a = root(a,1),b = root(b,1),c = root(c,1),true],
[a = root(a,1),root(b,1) < b,true,true],
[root(a,1) < a,true,c <= root(4*a*c-b^2,1),true]],
[[a],[b],[c,4*a*c-b^2],[a*x^2+b*x+c]]]
(%i6) [(qvpeds([all],[c,b,a,x],0,h1,r11,0),
qe(bfpcad(ext('(x^4+a*x^2+b*x+c >= 0))))),%pp]
Evaluation took 2.7640 seconds (2.7610 elapsed) using 665.230 MB.
(%o6) [[[c = root(c,1),b = root(126976*c^3-135*b^4,1),root(a,1) <= a,true],
[root(c,1) < c,b < root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true],
[root(c,1) < c,b = root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,1)
<= a,true],
[root(c,1) < c,root(b,1) < b,
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true]],
[[c],
[b,256*c^3-27*b^4,1024*c^3-2187*b^4,1024*c^3+3*b^4,4096*c^3+27*b^4,
126976*c^3-135*b^4],
[a,8*a*c-9*b^2-2*a^3,
256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2],
[x^4+a*x^2+b*x+c]]]
(%i7) [(qvpeds([all],[c,b,a,x],0,l,r11,0),
qe(bfpcad(ext('(x^4+a*x^2+b*x+c >= 0))))),%pp]
Evaluation took 0.7110 seconds (0.7090 elapsed) using 174.679 MB.
(%o7) [[[c = root(c,1),b = root(4096*c^3+27*b^4,1),root(a,1) <= a,true],
[root(c,1) < c,b < root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true],
[root(c,1) < c,b = root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,1)
<= a,true],
[root(c,1) < c,root(b,1) < b,
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true]],
[[c],[b,256*c^3-27*b^4,4096*c^3+27*b^4],
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2],
[x^4+a*x^2+b*x+c]]]
(%i8) [(qvpeds([all],[d,c,a,b,x],0,h1,r11,0),
qe(bfpcad(ext('(a*x^4+b*x^2+c*x+d >= 0))))),%pp]
Evaluation took 5.4500 seconds (5.4580 elapsed) using 1195.996 MB.
(%o8) [[[d = root(d,1),c = root(c,1),root(a,1) <= a,root(b,1) <= b,true],
[root(d,1) < d,c < root(c,1),root(a,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,2)
<= b,true],
[root(d,1) < d,c = root(c,1),root(126976*a*d^3-135*c^4,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,1)
<= b,true],
[root(d,1) < d,root(c,1) < c,root(a,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,2)
<= b,true]],
[[d],[c],
[a,256*a*d^3-27*c^4,1024*a*d^3-2187*c^4,1024*a*d^3+3*c^4,
4096*a*d^3+27*c^4,126976*a*d^3-135*c^4],
[b,8*a*b*d-9*a*c^2-2*b^3,
256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4-4*b^3*c^2],
[a*x^4+b*x^2+c*x+d]]]
(%i9) [(qvpeds([all],[d,c,a,b,x],0,l,r11,0),
qe(bfpcad(ext('(a*x^4+b*x^2+c*x+d >= 0))))),%pp]
Evaluation took 1.5370 seconds (1.5370 elapsed) using 368.276 MB.
(%o9) [[[d = root(d,1),c = root(c,1),root(a,1) <= a,root(b,1) <= b,true],
[root(d,1) < d,c < root(c,1),root(a,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,2)
<= b,true],
[root(d,1) < d,c = root(c,1),root(4096*a*d^3+27*c^4,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,1)
<= b,true],
[root(d,1) < d,root(c,1) < c,root(a,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,2)
<= b,true]],
[[d],[c],[a,256*a*d^3-27*c^4,4096*a*d^3+27*c^4],
[256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4-4*b^3*c^2],
[a*x^4+b*x^2+c*x+d]]]
(%i10) [(qvpeds([],[x,z,y],0,h1,r11,0),
qe(bfpcad(ext('(z^2+y^2+x^2-1 = 0 and z^3+x*z+y = 0))))),%pp]
Evaluation took 0.2330 seconds (0.2330 elapsed) using 44.868 MB.
(%o10) [[[x = root(x+1,1),z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,1),
y = root(z^3+x*z+y,1)],
[root(x+1,1) < x and x < root(x-1,1),
z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,1)
or z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,2),
y = root(z^3+x*z+y,1)],
[x = root(x-1,1),z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,1),
y = root(z^3+x*z+y,1)]],
[[x-1,x,x+1,x^2-3,3*x^3-2*x^2-3*x-2,
4*x^5-31*x^4+32*x^3+46*x^2-36*x-31],[z^6+2*x*z^4+(x^2+1)*z^2+x^2-1],
[z^3+x*z+y]]]
(%i11) [(qvpeds([],[x,z,y],0,l,r11,0),
qe(bfpcad(ext('(z^2+y^2+x^2-1 = 0 and z^3+x*z+y = 0))))),%pp]
Evaluation took 0.1250 seconds (0.1300 elapsed) using 22.110 MB.
(%o11) [[[x = root(x+1,1),z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,1),
y = root(z^3+x*z+y,1)],
[root(x+1,1) < x and x < root(x-1,1),
z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,1)
or z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,2),
y = root(z^3+x*z+y,1)],
[x = root(x-1,1),z = root(z^6+2*x*z^4+(x^2+1)*z^2+x^2-1,1),
y = root(z^3+x*z+y,1)]],
[[x-1,x+1,4*x^5-31*x^4+32*x^3+46*x^2-36*x-31],
[z^6+2*x*z^4+(x^2+1)*z^2+x^2-1],[z^3+x*z+y]]]
(%i12) [(qvpeds([],[y,x],0,h1,r11,0),
qe(bfpcad(ext('(y^4-2*y^3+y^2+(-3)*x^2*y+2*x^4 = 0))))),%pp]
Evaluation took 0.1530 seconds (0.1530 elapsed) using 20.637 MB.
(%o12) [[[y = root(y,1),x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)],
[root(y,1) < y and y < root(y-1,1),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,3)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,4)],
[y = root(y-1,1),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,3)],
[root(y-1,1) < y and y < root(8*y^2-16*y-1,2),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,3)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,4)],
[y = root(8*y^2-16*y-1,2),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)]],
[[y-1,y,8*y^2-16*y-1],[y^4-2*y^3+y^2-3*x^2*y+2*x^4]]]
(%i13) [(qvpeds([],[y,x],0,l,r11,0),
qe(bfpcad(ext('(y^4-2*y^3+y^2+(-3)*x^2*y+2*x^4 = 0))))),%pp]
Evaluation took 0.1170 seconds (0.1170 elapsed) using 20.605 MB.
(%o13) [[[y = root(y,1),x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)],
[root(y,1) < y and y < root(y-1,1),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,3)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,4)],
[y = root(y-1,1),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,3)],
[root(y-1,1) < y and y < root(8*y^2-16*y-1,2),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,3)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,4)],
[y = root(8*y^2-16*y-1,2),
x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,1)
or x = root(y^4-2*y^3+y^2-3*x^2*y+2*x^4,2)]],
[[y-1,y,8*y^2-16*y-1],[y^4-2*y^3+y^2-3*x^2*y+2*x^4]]]
(%i14) [(qvpeds([],[y,x],0,h1,r11,0),
qe(bfpcad(ext('(144*y^2+96*x^2*y+9*x^4+105*x^2+70*x-98 = 0
and x*y^2+6*x*y+x^3+9*x = 0))))),%pp]
Evaluation took 0.0780 seconds (0.0780 elapsed) using 9.584 MB.
(%o14) [[[y = root(72*y^2-49,1) or y = root(72*y^2-49,2),
x = root(1132535906187264*y^9+2648683072534464*y^8
-11812443576362496*y^7
-144938986561480320*y^6
-169128077155358976*y^5
+764415088047207513*y^4
+4188447484701833676*y^3
+3270406842477283485*y^2
-2768658167053333674*y
+419177363858840230*x
-2534619807825082586,1)]],
[[72*y^2-49],
[1132535906187264*y^9+2648683072534464*y^8-11812443576362496*y^7
-144938986561480320*y^6-169128077155358976*y^5
+764415088047207513*y^4+4188447484701833676*y^3
+3270406842477283485*y^2-2768658167053333674*y
+419177363858840230*x-2534619807825082586]]]
(%i15) [(qvpeds([],[y,x],0,l,r11,0),
qe(bfpcad(ext('(144*y^2+96*x^2*y+9*x^4+105*x^2+70*x-98 = 0
and x*y^2+6*x*y+x^3+9*x = 0))))),%pp]
Evaluation took 0.0780 seconds (0.0800 elapsed) using 9.506 MB.
(%o15) [[[y = root(72*y^2-49,1) or y = root(72*y^2-49,2),
x = root(1132535906187264*y^9+2648683072534464*y^8
-11812443576362496*y^7
-144938986561480320*y^6
-169128077155358976*y^5
+764415088047207513*y^4
+4188447484701833676*y^3
+3270406842477283485*y^2
-2768658167053333674*y
+419177363858840230*x
-2534619807825082586,1)]],
[[72*y^2-49],
[1132535906187264*y^9+2648683072534464*y^8-11812443576362496*y^7
-144938986561480320*y^6-169128077155358976*y^5
+764415088047207513*y^4+4188447484701833676*y^3
+3270406842477283485*y^2-2768658167053333674*y
+419177363858840230*x-2534619807825082586]]]
(%i16) [(qvpeds([ex,ex],[x,y,z,a,b],0,h1,r11,0),
qe(bfpcad(ext('(x-a*b = 0 and y-a*b^2 = 0 and z-a^2 = 0))))),%pp]
Evaluation took 0.4380 seconds (0.4390 elapsed) using 97.523 MB.
(%o16) [[[x < root(x,1),y < root(y,1) or root(y,1) < y,z = root(y^2*z-x^4,1),
true,true],[x = root(x,1),y = root(y,1),root(z,1) <= z,true,true],
[root(x,1) < x,y < root(y,1) or root(y,1) < y,z = root(y^2*z-x^4,1),
true,true]],
[[x],[y],[z,z*y^2-x^4],[a,a*y-x^2,(-a^2)+z,z*y-a*x^2],
[x-a*b,y-b*x,(-a*x)+z*b]]]
(%i17) [(qvpeds([ex,ex],[x,y,z,a,b],0,l,r11,0),
qe(bfpcad(ext('(x-a*b = 0 and y-a*b^2 = 0 and z-a^2 = 0))))),%pp]
Evaluation took 0.4270 seconds (0.4270 elapsed) using 96.663 MB.
(%o17) [[[x < root(x,1),y < root(y,1) or root(y,1) < y,z = root(y^2*z-x^4,1),
true,true],[x = root(x,1),y = root(y,1),root(z,1) <= z,true,true],
[root(x,1) < x,y < root(y,1) or root(y,1) < y,z = root(y^2*z-x^4,1),
true,true]],
[[x],[y],[z,(-x^4)+z*y^2],[a,(-x^2)+y*a,(-a^2)+z,(-a*x^2)+z*y],
[x-a*b,(-b*x)+y,(-a*x)+z*b]]]
(%i18) [(qvpeds([ex,ex,ex],[x,y,z],0,h1,r11,0),
qe(bfpcad(ext('(x^2+y^2+z^2-1 < 0 and x^2+(y+z-2)^2-1 < 0))))),%pp]
Evaluation took 0.3330 seconds (0.3330 elapsed) using 73.647 MB.
(%o18) [[[true,true,true]],
[[x-1,x,x+1,x^4+22*x^2-7],
[y^2+x^2-1,y^4-4*y^3+(x^2+7)*y^2+((-4*x^2)-4)*y+4*x^2],
[z^2+y^2+x^2-1,z^2+(2*y-4)*z+y^2-4*y+x^2+3]]]
(%i19) [(qvpeds([ex,ex,ex],[x,y,z],0,l,r11,0),
qe(bfpcad(ext('(x^2+y^2+z^2-1 < 0 and x^2+(y+z-2)^2-1 < 0))))),%pp]
Evaluation took 0.3160 seconds (0.3170 elapsed) using 65.861 MB.
(%o19) [[[true,true,true]],
[[x-1,x,x+1,x^4+22*x^2-7],
[y^2+x^2-1,y^4-4*y^3+(x^2+7)*y^2+((-4*x^2)-4)*y+4*x^2],
[z^2+y^2+x^2-1,z^2+(2*y-4)*z+y^2-4*y+x^2+3]]]
(%i20) [(qvpeds([ex,all,all],[r,x,y],0,h1,r11,0),
qe(bfpcad(ext('(x-r > 0 and y-r > 0
implies x^2*(1+2*y)^2-y^2*(1+2*x^2) > 0))))),%pp]
Evaluation took 0.9550 seconds (0.9550 elapsed) using 189.163 MB.
(%o20) [[[true,true,true]],
[[r,r+2,2*r+1,4*r+1,2*r^2-1,2*r^2+4*r+1],
[x,x-r,2*x^2-1,2*x^2+1,(2*r^2+4*r+1)*x^2-r^2],
[y-r,(2*x^2-1)*y^2+4*x^2*y+x^2]]]
(%i21) [(qvpeds([ex,all,all],[r,x,y],0,l,r11,0),
qe(bfpcad(ext('(x-r > 0 and y-r > 0
implies x^2*(1+2*y)^2-y^2*(1+2*x^2) > 0))))),%pp]
Evaluation took 0.9320 seconds (0.9330 elapsed) using 175.511 MB.
(%o21) [[[true,true,true]],
[[r,r+2,2*r+1,4*r+1,2*r^2-1,2*r^2+4*r+1],
[x,x-r,2*x^2-1,2*x^2+1,(2*r^2+4*r+1)*x^2-r^2],
[y-r,(2*x^2-1)*y^2+4*x^2*y+x^2]]]
(%i22) [(qvpeds([ex],[a,b,r],0,h1r,r11,0),
qe(bfpcad(ext('(3*a^2*r+3*b^2+(-2)*a*r-a^2-b^2 < 0
and 3*a^2*r+3*b^2*r+(-4)*a*r+r+(-2)*a^2+(-2)*b^2+2*a
> 0 and a-1/2 >= 0 and b > 0 and r > 0
and r-1 < 0))))),%pp]
Evaluation took 2.5190 seconds (2.5200 elapsed) using 659.301 MB.
(%o22) [[[root(2*a-1,1) <= a and a < root(3*a-2,1),
root(b,1) < b and b < root(b^2+a^2-a,2),true],
[a = root(3*a-2,1),root(b,1) < b and b <= root(b^2+a^2-a,2),true],
[root(3*a-2,1) < a and a < root(a-1,1),
root(b,1) < b and b < root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3
+3*a^2,4),true],
[a = root(a-1,1),
root(b,1) < b and b < root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3
+3*a^2,3),true],
[root(a-1,1) < a and a < root(9*a^4-72*a^3+108*a^2-48*a+4,3),
root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,3) < b
and b < root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,4),true]],
[[9*a^4-72*a^3+108*a^2-48*a+4,9*a^2-12*a+2,3*a-1,3*a-2,2*a-1,a,a-1],
[6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,3*b^2+3*a^2-4*a+1,
2*b^2-a^2,b^2+a^2-a,b^2+a^2-2*a+1,b],
[(3*b^2+3*a^2-4*a+1)*r-2*b^2-2*a^2+2*a,(3*a^2-2*a)*r+2*b^2-a^2,r,
r-1]]]
(%i23) [(qvpeds([ex],[a,b,r],0,l,r11,0),
qe(bfpcad(ext('(3*a^2*r+3*b^2+(-2)*a*r-a^2-b^2 < 0
and 3*a^2*r+3*b^2*r+(-4)*a*r+r+(-2)*a^2+(-2)*b^2+2*a
> 0 and a-1/2 >= 0 and b > 0 and r > 0
and r-1 < 0))))),%pp]
Evaluation took 2.1620 seconds (2.1620 elapsed) using 514.650 MB.
(%o23) [[[root(2*a-1,1) <= a and a < root(3*a-2,1),
root(b,1) < b and b < root(b^2+a^2-a,2),true],
[a = root(3*a-2,1),root(b,1) < b and b < root(2*b^2-a^2,2),true],
[root(3*a-2,1) < a and a < root(a-1,1),
root(b,1) < b and b < root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3
+3*a^2,4),true],
[a = root(a-1,1),
root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,2) < b
and b < root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,3),true],
[root(a-1,1) < a and a < root(9*a^4-72*a^3+108*a^2-48*a+4,3),
root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,3) < b
and b < root(6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2,4),true]],
[[a-1,a,2*a-1,3*a-2,3*a-1,9*a^4-72*a^3+108*a^2-48*a+4],
[b,b^2+a^2-2*a+1,b^2+a^2-a,2*b^2-a^2,3*b^2+3*a^2-4*a+1,
6*b^4+(9*a^2-12*a+2)*b^2+3*a^4-6*a^3+3*a^2],
[r-1,r,(3*a^2-2*a)*r+2*b^2-a^2,
(3*b^2+3*a^2-4*a+1)*r-2*b^2-2*a^2+2*a]]]
(%i24) [(qvpeds([all,all,all,ex],[y,x,a,b,c,z],0,h1,r11,0),
qe(bfpcad(ext('(a > 0 and a*z^2+b*z+c # 0
implies a*x^2+b*x+c-y > 0))))),%pp]
Evaluation took 3.0010 seconds (3.0030 elapsed) using 622.586 MB.
(%o24) [[[y <= root(y,1),true,true,true,true,true]],
[[y],[x],[a,(-x^2*a)+y],
[b,(-x*b)-x^2*a+y,(-b^2)-4*x*a*b-4*x^2*a^2+4*y*a],
[c,4*a*c-b^2,(-c)-x*b-x^2*a+y],[a*z^2+b*z+c]]]
(%i25) [(qvpeds([all,all,all,ex],[y,x,a,b,c,z],0,l,r11,0),
qe(bfpcad(ext('(a > 0 and a*z^2+b*z+c # 0
implies a*x^2+b*x+c-y > 0))))),%pp]
Evaluation took 2.9450 seconds (2.9450 elapsed) using 598.900 MB.
(%o25) [[[y <= root(y,1),true,true,true,true,true]],
[[y],[x],[a,(-x^2*a)+y],
[b,(-x*b)-x^2*a+y,(-b^2)-4*x*a*b-4*x^2*a^2+4*y*a],
[c,4*a*c-b^2,(-c)-x*b-x^2*a+y],[a*z^2+b*z+c]]]
(%i26) [(qvpeds([all,all],[b,a,c,x,y],0,h1r,r11,0),
qe(bfpcad(ext('(0 < a and 0 < b
and ((x-c)^2/a^2+y^2/b^2 = 1
implies x^2+y^2 <= 1)))))),%pp]
Evaluation took 76.6110 seconds (76.6430 elapsed) using 15531.535 MB.
(%o26) [[[root(b,1) < b and b < root(b-1,1),
root(a,1) < a and a < root(b^2-a,1),
root(b^2*c^2+b^4+((-a^2)-1)*b^2+a^2,1) <= c
and c <= root(b^2*c^2+b^4+((-a^2)-1)*b^2+a^2,2),true,true],
[root(b,1) < b and b < root(b-1,1),
root(b^2-a,1) <= a and a < root(a-1,1),
root(c-a+1,1) <= c and c <= root(c+a-1,1),true,true],
[root(b,1) < b and b < root(b-1,1),a = root(a-1,1),c = root(c,1),
true,true],
[b = root(b-1,1),root(a,1) < a and a <= root(a-1,1),c = root(c,1),
true,true]],
[[b+1,b,b-1],
[(2*a+1)*b^2+a^2,(2*a-1)*b^2-a^2,b^2+a,b^2-a,b+a,b-a,a+1,a,a-1],
[b^2*c^2+b^4+((-a^2)-1)*b^2+a^2,b^2*c^2-a^2*b^2+a^2,c+a+1,c+a-1,
c-a+1,c-a-1,c],
[(b^2-a^2)*x^2-2*b^2*c*x+b^2*c^2-a^2*b^2+a^2,x-c+a,x-c-a,x+1,x-1],
[a^2*y^2+b^2*x^2-2*b^2*c*x+b^2*c^2-a^2*b^2,y^2+x^2-1]]]
(%i27) [(qvpeds([all,all],[b,a,c,x,y],0,l,r11,0),
qe(bfpcad(ext('(0 < a and 0 < b
and ((x-c)^2/a^2+y^2/b^2 = 1
implies x^2+y^2 <= 1)))))),%pp]
Evaluation took 62.6640 seconds (62.6860 elapsed) using 11842.095 MB.
(%o27) [[[root(b,1) < b and b < root(b-1,1),
root(a,1) < a and a <= root(b^2-a,1),
root(b^2*c^2+b^4+((-a^2)-1)*b^2+a^2,1) <= c
and c <= root(b^2*c^2+b^4+((-a^2)-1)*b^2+a^2,2),true,true],
[root(b,1) < b and b < root(b-1,1),
root(b^2-a,1) < a and a < root(a-1,1),
root(c-a+1,1) <= c and c <= root(c+a-1,1),true,true],
[root(b,1) < b and b < root(b-1,1),a = root(a-1,1),c = root(c+a-1,1),
true,true],
[b = root(b-1,1),root(a,1) < a and a <= root((2*a-1)*b^2-a^2,1),
c = root(b^2*c^2+b^4+((-a^2)-1)*b^2+a^2,1),true,true]],
[[b-1,b,b+1],
[a-1,a,a+1,b-a,b+a,b^2-a,b^2+a,(2*a-1)*b^2-a^2,(2*a+1)*b^2+a^2],
[c-a-1,c-a+1,c+a-1,c+a+1,b^2*c^2-a^2*b^2+a^2,
b^2*c^2+b^4+((-a^2)-1)*b^2+a^2],
[x-1,x+1,x-c-a,x-c+a,(b^2-a^2)*x^2-2*b^2*c*x+b^2*c^2-a^2*b^2+a^2],
[y^2+x^2-1,a^2*y^2+b^2*x^2-2*b^2*c*x+b^2*c^2-a^2*b^2]]]
(%i28) [(qvpeds([ex,all,all],[d,c,a,b],0,h1r,r11,0),
qe(bfpcad(ext('((a-d = 0 and b-c = 0 or a-c = 0 and b-1 = 0)
implies a^2-b = 0))))),%pp]
Evaluation took 9.7020 seconds (9.7060 elapsed) using 1360.997 MB.
(%o28) [[[d = root(d+1,1) or d = root(d-1,1),true,true,true]],
[[d+1,d,d-1],[(-c)+d^2,(-c)+d,c+1,c,c-1],
[(-a)+d,(-a^2)+c,(-a)+c,a+1,a-1],[(-b)+c,b-a^2,b-1]]]
(%i29) [(qvpeds([ex,all,all],[d,c,a,b],0,l,r11,0),
qe(bfpcad(ext('((a-d = 0 and b-c = 0 or a-c = 0 and b-1 = 0)
implies a^2-b = 0))))),%pp]
Evaluation took 9.5980 seconds (9.6010 elapsed) using 1325.779 MB.
(%o29) [[[d = root(d+1,1) or d = root(d-1,1),true,true,true]],
[[d-1,d,d+1],[c-1,c,c+1,(-c)+d,(-c)+d^2],
[a-1,a+1,(-a)+c,(-a^2)+c,(-a)+d],[b-1,b-a^2,(-b)+c]]]
(%i30) [(qvpeds([ex,ex,ex],[b,a,x,y,z],0,h1,r11,0),
qe(bfpcad(ext('(x+y+z = 0 and x*y+y*z+z*x-a = 0 and x*y*z-b = 0))))),
%pp]
Evaluation took 1.1200 seconds (1.1250 elapsed) using 273.509 MB.
(%o30) [[[true,a <= root(27*b^2+4*a^3,1),true,true,true]],
[[b],[a,4*a^3+27*b^2],[3*x^2+4*a,x^3+a*x-b],[y^2+x*y+x^2+a],[y+x+z]]]
(%i31) [(qvpeds([ex,ex,ex],[b,a,x,y,z],0,l,r11,0),
qe(bfpcad(ext('(x+y+z = 0 and x*y+y*z+z*x-a = 0 and x*y*z-b = 0))))),
%pp]
Evaluation took 0.7980 seconds (0.7980 elapsed) using 145.864 MB.
(%o31) [[[true,a <= root(27*b^2+4*a^3,1),true,true,true]],
[[b],[a,4*a^3+27*b^2],[3*x^2+4*a,x^3+a*x-b],[y^2+x*y+x^2+a],[y+x+z]]]
(%i32) [(qvpeds([ex,ex,ex],[a,b,c,x,y,z],0,h1r,r11,0),
qe(bfpcad(ext('(x+y+z = a and x*y+y*z+z*x = b and x*y*z = c))))),%pp]
Evaluation took 0.7660 seconds (0.7660 elapsed) using 184.901 MB.
(%o32) [[[true,b < root(3*b-a^2,1),
root(27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2,1) <= c
and c <= root(27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2,2),true,true,
true],
[true,b = root(3*b-a^2,1),
c = root(27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2,1),true,true,true]],
[[],[3*b-a^2],[27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2],
[x^3-a*x^2+b*x-c,3*x^2-2*a*x+4*b-a^2],[y^2+x*y-a*y+x^2-a*x+b],
[z+y+x-a]]]
(%i33) [(qvpeds([ex,ex,ex],[a,b,c,x,y,z],0,l,r11,0),
qe(bfpcad(ext('(x+y+z = a and x*y+y*z+z*x = b and x*y*z = c))))),%pp]
Evaluation took 0.6550 seconds (0.6560 elapsed) using 140.720 MB.
(%o33) [[[true,b < root(3*b-a^2,1),
root(27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2,1) <= c
and c <= root(27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2,2),true,true,
true],
[true,b = root(3*b-a^2,1),
c = root(27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2,1),true,true,true]],
[[],[3*b-a^2],[27*c^2-18*a*b*c+4*a^3*c+4*b^3-a^2*b^2],
[3*x^2-2*a*x+4*b-a^2,x^3-a*x^2+b*x-c],[y^2+x*y-a*y+x^2-a*x+b],
[z+y+x-a]]]
(%i34) [(qvpeds([ex,ex,ex],[t,x,y],0,h1r,r11,0),
qe(bfpcad(ext('((17/16)*t-6 >= 0 and (17/16)*t-10 <= 0
and x-(17/16)*t+1 >= 0
and x-(17/16)*t-1 <= 0
and y-(17/16)*t+9 >= 0
and y-(17/16)*t+7 <= 0
and (x-t)^2+y^2-1 <= 0))))),%pp]
Evaluation took 3.1300 seconds (3.1330 elapsed) using 890.252 MB.
(%o34) [[[true,true,true]],
[[145*t^2-1920*t+6272,145*t^2-2464*t+10368,17*t-96,17*t-112,17*t-128,
17*t-144,17*t-160,t+32,t,t-32],
[256*x^2-512*t*x+545*t^2-3808*t+12288,
256*x^2-512*t*x+545*t^2-4896*t+20480,16*x-17*t+16,16*x-17*t-16,
x-t+1,x-t-1],[y^2+x^2-2*t*x+t^2-1,16*y-17*t+144,16*y-17*t+112]]]
(%i35) [(qvpeds([ex,ex,ex],[t,x,y],0,l,r11,0),
qe(bfpcad(ext('((17/16)*t-6 >= 0 and (17/16)*t-10 <= 0
and x-(17/16)*t+1 >= 0
and x-(17/16)*t-1 <= 0
and y-(17/16)*t+9 >= 0
and y-(17/16)*t+7 <= 0
and (x-t)^2+y^2-1 <= 0))))),%pp]
Evaluation took 1.5770 seconds (1.5780 elapsed) using 246.028 MB.
(%o35) [[[true,true,true]],
[[t-32,t,t+32,17*t-160,17*t-144,17*t-128,17*t-112,17*t-96,
145*t^2-2464*t+10368,145*t^2-1920*t+6272],
[x-t-1,x-t+1,16*x-17*t-16,16*x-17*t+16,
256*x^2-512*t*x+545*t^2-4896*t+20480,
256*x^2-512*t*x+545*t^2-3808*t+12288],
[16*y-17*t+112,16*y-17*t+144,y^2+x^2-2*t*x+t^2-1]]]
(%i36) [fpprec,%ez,ratepsilon]:[48,1.0b-12,1.0b-48]
Evaluation took 0.0010 seconds (0.0000 elapsed) using 0 bytes.
(%o36) [48,1.0b-12,1.0b-48]
(%i37) f0:x < 0 and 50000*(x^2+y^2)-49719 < 0
implies 360000+720000*x+720000*x^2+480000*x^3+240000*x^4+96000*x^5
+32200*x^6+9200*x^7+2225*x^8+450*x^9+75*x^10+10*x^11
+x^12+(-3000)*x^4*y^2+1200*x^5*y^2+2100*x^6*y^2
+1000*x^7*y^2+275*x^8*y^2+50*x^9*y^2+6*x^10*y^2
+3000*x^2*y^4+(-6000)*x^3*y^4+(-2250)*x^4*y^4
+300*x^5*y^4+350*x^6*y^4+100*x^7*y^4+15*x^8*y^4
+(-200)*y^6+2000*x*y^6+(-1900)*x^2*y^6+(-600)*x^3*y^6
+150*x^4*y^6+100*x^5*y^6+20*x^6*y^6+225*y^8
+(-350)*x*y^8+(-25)*x^2*y^8+50*x^3*y^8+15*x^4*y^8
+(-25)*y^10+10*x*y^10+6*x^2*y^10+y^12-360000
< 0
Evaluation took 0.0290 seconds (0.0300 elapsed) using 3.558 MB.
(%i38) [(qvpeds([all,all],[x,y],0,h1,r11,0),qe(bfpcad(ext(f0)))),%pp]
Evaluation took 10.0710 seconds (10.0740 elapsed) using 2590.940 MB.
(%o38) [[[true,true]],
[[x,6*x^2+10*x-25,28*x^2-68*x-17,50000*x^2-49719,
x^5+5*x^4+25*x^3+100*x^2+300*x+600,
28*x^6-204*x^5-101*x^4-190*x^3+5269*x^2-5710*x+1875,
1088*x^6-7552*x^5+12736*x^4-3600*x^3+21000*x^2-41240*x+14039,
200000000000000000000000000000000*x^6
+599438000000000000000000000000000*x^5
+1225000789610000000000000000000000*x^4
+2442297336707065300000000000000000*x^3
+3754710227977240336626750000000000*x^2
+3758510381317956468954765766500000*x-5402844825834997735210773,
1088*x^12-15104*x^11+59904*x^10-3344*x^9+112936*x^8-298200*x^7
-6467281*x^6+17310870*x^5-5750825*x^4+6290450*x^3
-14463075*x^2+6471500*x-580150,
120832*x^12-989184*x^11+1965568*x^10+1699840*x^9+3373952*x^8
-10134400*x^7-57194464*x^6+73892480*x^5-12046600*x^4
-5395400*x^3-126535850*x^2+56676900*x-290075,
120832*x^19-1648640*x^18+5940736*x^17+8692736*x^16-23252608*x^15
-246864000*x^14-1190564192*x^13-3079579840*x^12
+12973976600*x^11+1232865000*x^10-84076534050*x^9
-209944983400*x^8+39826614875*x^7+1457720531250*x^6
+2010504800625*x^5+25224378750*x^4-957912804375*x^3
-949924375000*x^2+750565675000*x-43511250000,
4571136*x^19+2818048*x^18-25075712*x^17+34385920*x^16+930661376*x^15
+3433646080*x^14+8198400*x^13-47876089600*x^12
-184510347200*x^11-302452144000*x^10+194216259600*x^9
+2048648030000*x^8+4813290745000*x^7+5892848620000*x^6
+2971088911875*x^5-473950680000*x^4-1572986706250*x^3
+258053312500*x^2+440250359375*x+1631671875,
4571136*x^28+4227072*x^27-51011584*x^26+208793600*x^25
+4487152640*x^24+21158707200*x^23-2748128000*x^22
-513743699200*x^21-1859278084800*x^20
+11836781184000*x^19+177227571163600*x^18
+1147497743010000*x^17+5032673524115000*x^16
+16422964823940000*x^15+40583034110816875*x^14
+73030624007188750*x^13+78801621529646875*x^12
-18223494396843750*x^11-288264068603906250*x^10
-681410411614046875*x^9-946722681541984375*x^8
-792370510167421875*x^7-273080203974921875*x^6
+181512821587890625*x^5+197555235933593750*x^4
-7722981467968750*x^3-126478640906250000*x^2
-6832921593750000*x-1599038437500000,
855113728*x^28+21377843200*x^27+283256422400*x^26+2608096870400*x^25
+18521228902400*x^24+106982744064000*x^23
+518383694233600*x^22+2147375910912000*x^21
+7692855189760000*x^20+23980542444800000*x^19
+65142992800320000*x^18+153700917730560000*x^17
+311998085014880000*x^16+534257219535200000*x^15
+740482905349000000*x^14+746555405314000000*x^13
+321582900808375000*x^12-619086555599625000*x^11
-1781201761577531250*x^10-2496929229230000000*x^9
-2168728237020312500*x^8-905392388410156250*x^7
+397713892669921875*x^6+915322655607031250*x^5
+609971037097656250*x^4+165439917011718750*x^3
-20690934755859375*x^2+38769456621093750*x
-1019794921875],
[50000*y^2+50000*x^2-49719,
y^12+(6*x^2+10*x-25)*y^10+(15*x^4+50*x^3-25*x^2-350*x+225)*y^8
+(20*x^6+100*x^5+150*x^4-600*x^3-1900*x^2+2000*x-200)*y^6
+(15*x^8+100*x^7+350*x^6+300*x^5-2250*x^4-6000*x^3+3000*x^2)*y^4
+(6*x^10+50*x^9+275*x^8+1000*x^7+2100*x^6+1200*x^5-3000*x^4)*y^2
+x^12+10*x^11+75*x^10+450*x^9+2225*x^8+9200*x^7+32200*x^6
+96000*x^5+240000*x^4+480000*x^3+720000*x^2+720000*x]]]
(%i39) [(qvpeds([all,all],[x,y],0,l,r11,0),qe(bfpcad(ext(f0)))),%pp]
Evaluation took 1.6160 seconds (1.6180 elapsed) using 603.005 MB.
(%o39) [[[true,true]],
[[x,50000*x^2-49719,x^5+5*x^4+25*x^3+100*x^2+300*x+600,
200000000000000000000000000000000*x^6
+599438000000000000000000000000000*x^5
+1225000789610000000000000000000000*x^4
+2442297336707065300000000000000000*x^3
+3754710227977240336626750000000000*x^2
+3758510381317956468954765766500000*x-5402844825834997735210773,
855113728*x^28+21377843200*x^27+283256422400*x^26+2608096870400*x^25
+18521228902400*x^24+106982744064000*x^23
+518383694233600*x^22+2147375910912000*x^21
+7692855189760000*x^20+23980542444800000*x^19
+65142992800320000*x^18+153700917730560000*x^17
+311998085014880000*x^16+534257219535200000*x^15
+740482905349000000*x^14+746555405314000000*x^13
+321582900808375000*x^12-619086555599625000*x^11
-1781201761577531250*x^10-2496929229230000000*x^9
-2168728237020312500*x^8-905392388410156250*x^7
+397713892669921875*x^6+915322655607031250*x^5
+609971037097656250*x^4+165439917011718750*x^3
-20690934755859375*x^2+38769456621093750*x
-1019794921875],
[50000*y^2+50000*x^2-49719,
y^12+(6*x^2+10*x-25)*y^10+(15*x^4+50*x^3-25*x^2-350*x+225)*y^8
+(20*x^6+100*x^5+150*x^4-600*x^3-1900*x^2+2000*x-200)*y^6
+(15*x^8+100*x^7+350*x^6+300*x^5-2250*x^4-6000*x^3+3000*x^2)*y^4
+(6*x^10+50*x^9+275*x^8+1000*x^7+2100*x^6+1200*x^5-3000*x^4)*y^2
+x^12+10*x^11+75*x^10+450*x^9+2225*x^8+9200*x^7+32200*x^6
+96000*x^5+240000*x^4+480000*x^3+720000*x^2+720000*x]]]
以下の出力は長い為，省略しました．
(%i40) [fpprec,%ez,ratepsilon]: [32,1.0b-6,1.0b-32];
Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
(%o40) [32,1.0b-6,1.0b-32]
(%i41) [(qvpeds ([],[x,y,z],0,h1,r11,0 ),
qe( bfpcad(ext( '( (y-1)*z^4+x*z^3+x*(1-y)*z^2+(y-x-1)*z+y=0 ) ))),%pp)];
Evaluation took 163.0310 seconds (163.1330 elapsed) using 57799.604 MB.
(%o41) [[[x-6,x,x+1,x+2,3*x+8,8*x+3,4*x^2-27,2*x^3-8*x-9,2*x^3-24*x^2+72*x+27,
8*x^3+7*x^2-42*x-36,x^4+38*x^3+18*x^2-156*x-108,
2*x^4+47*x^3-124*x-72,4*x^4+x^3-16*x^2-22*x-6,
4*x^4+21*x^3-4*x^2-36*x-18,5*x^4+26*x^3-9*x^2-54*x-27,
8*x^4+96*x^3-416*x^2+201*x+282,16*x^4-95*x^3+24*x^2+704*x+512,
16*x^4+4*x^3-128*x^2-144*x+229,44*x^4+52*x^3-391*x^2-318*x+303,
4*x^5-88*x^4+645*x^3-1854*x^2+1920*x-1024,
8*x^5+281*x^4+146*x^3-1806*x^2-1404*x+363,
15*x^5+278*x^4-30*x^3-1171*x^2-972*x-142,x^6-4*x^5+7*x^3-x^2-3*x+1,
x^6+46*x^5+142*x^4-74*x^3-426*x^2-342*x-81,
x^6+78*x^5+556*x^4-208*x^3-1898*x^2-1608*x-405,
16*x^6+4*x^5-128*x^4-272*x^3+85*x^2+485*x+256,
192*x^9+1952*x^8-7786*x^7-18888*x^6+41108*x^5+45845*x^4+11523*x^3
-47238*x^2-78912*x-104832,
x^10-36*x^9+540*x^8-4212*x^7+16875*x^6-23679*x^5-43659*x^4+114669*x^3
+96228*x^2-58320*x-46656,
4*x^11-24*x^10-640*x^9+8377*x^8-38996*x^7+57424*x^6+130396*x^5
-397672*x^4-88296*x^3+476802*x^2+291240*x+52488,
2304*x^11-68736*x^10+753424*x^9-2747520*x^8-12830688*x^7
+156357720*x^6-532952352*x^5+497201760*x^4+874470897*x^3
-1352650752*x^2-339738624*x+905969664,
192*x^13-1408*x^12-7536*x^11+72256*x^10-85432*x^9-418072*x^8
+1080992*x^7+265376*x^6-3323600*x^5+1586496*x^4+3750759*x^3
-1186326*x^2-1959552*x-373248,
16*x^15-640*x^14+11060*x^13-106516*x^12+610680*x^11-1996383*x^10
+2610948*x^9+4680808*x^8-20830994*x^7+14014728*x^6
+32414544*x^5-20223465*x^4-40399848*x^3-20947032*x^2-5792544*x
-944784,
4096*x^19-76032*x^18-122880*x^17+10690448*x^16-84608248*x^15
+268079584*x^14-170213648*x^13-1481638472*x^12
+4738394744*x^11-1497816385*x^10-17768059852*x^9
+18831296016*x^8+43729432710*x^7-32989362828*x^6
-76761910980*x^5+8582283771*x^4+65151311040*x^3
+16307847936*x^2-14837686272*x-5792808960,
4096*x^20-163840*x^19+2852864*x^18-27393408*x^17+149542144*x^16
-388286848*x^15-214815103*x^14+4154362488*x^13
-7635762592*x^12-10267177106*x^11+44575897180*x^10
-943203000*x^9-117037926820*x^8+32371882616*x^7
+192201053844*x^6-23391280800*x^5-205468815744*x^4
-49971769344*x^3+106333876224*x^2+77778911232*x+16052649984,
1024*x^25-36864*x^24+258048*x^23+9652320*x^22-275282941*x^21
+3486936746*x^20-25967275698*x^19+114334396176*x^18
-227290439617*x^17-360243822520*x^16+3107900678073*x^15
-4769190186712*x^14-9207670651692*x^13+33891866135830*x^12
+3248200641279*x^11-103251848335494*x^10+24822591383424*x^9
+196171087421376*x^8-26483105274624*x^7-245526417483264*x^6
-55109356535808*x^5+147054424817664*x^4+90730087317504*x^3
-5655119265792*x^2-16731447754752*x-3522410053632,
63488*x^29-3822208*x^28+101436112*x^27-1558519760*x^26
+15218355663*x^25-96366411132*x^24+374174103239*x^23
-650077067955*x^22-1132725702993*x^21+8060985767681*x^20
-9709931306620*x^19-26781342092492*x^18+67248318526264*x^17
+73969079197117*x^16-283280225707950*x^15
-213264871269207*x^14+938560561534635*x^13
+452329212683124*x^12-1991414684849397*x^11
-890275611320538*x^10+2384378846393901*x^9
+1407423445298100*x^8-1270657944171792*x^7
-1145677558363200*x^6+23176788277248*x^5
+301610526289920*x^4+117605153636352*x^3-9039935176704*x^2
-19813556551680*x-3962711310336,
7936*x^36-758304*x^35+33419327*x^34-901346428*x^33+16631964287*x^32
-222094789346*x^31+2209871734238*x^30-16568887999552*x^29
+92997245415767*x^28-377363024540523*x^27
+989223612173741*x^26-867722678128324*x^25
-5048534162562257*x^24+24309955173059337*x^23
-41581904973045947*x^22-28328921944006469*x^21
+277043222823124631*x^20-427140634741112450*x^19
-394584630361333315*x^18+2108413813884902713*x^17
-1146104394578571495*x^16-4964331326135378292*x^15
+6113780241822578916*x^14+7684481244922767486*x^13
-13180637957355086364*x^12-8973842356231860348*x^11
+17034125906519891856*x^10+7340829016726477440*x^9
-16471368758915006976*x^8-5867495981300284416*x^7
+12273511269086613504*x^6+7406908509570859008*x^5
-2449053403743780864*x^4-2741599838700306432*x^3
-226468951385702400*x^2+292162779488452608*x
+68475651442606080],
[y-1,y,y-x-1,8*y^2-16*y+3*x+8,
(4*x^3-16*x-18)*y^4+((-16*x^3)-14*x^2+84*x+72)*y^3
+(x^4+38*x^3+18*x^2-156*x-108)*y^2
+((-2*x^4)-47*x^3+124*x+72)*y+4*x^4+21*x^3-4*x^2
-36*x-18,
(16*x^4+4*x^3-128*x^2-144*x+229)*y^6
+((-88*x^4)-104*x^3+782*x^2+636*x-606)*y^5
+(8*x^5+281*x^4+146*x^3-1806*x^2-1404*x+363)*y^4
+((-30*x^5)-556*x^4+60*x^3+2342*x^2+1944*x+284)*y^3
+(x^6+78*x^5+556*x^4-208*x^3-1898*x^2-1608*x-405)*y^2
+((-2*x^6)-92*x^5-284*x^4+148*x^3+852*x^2+684*x+162)*y+5*x^6+36*x^5
+48*x^4-46*x^3-144*x^2-108*x-27],
[(y-1)*z^4+x*z^3+((-x*y)+x)*z^2+(y-x-1)*z+y]]]
(%i42) [(qvpeds ([],[x,y,z],0,l,r11,0 ),
qe( bfpcad(ext( '( (y-1)*z^4+x*z^3+x*(1-y)*z^2+(y-x-1)*z+y=0 ) ))),%pp)];
Evaluation took 8.0460 seconds (8.0510 elapsed) using 3347.636 MB.
(%o42) [[[x,x+1,4*x^2-27,5*x^4+26*x^3-9*x^2-54*x-27,
16*x^4-95*x^3+24*x^2+704*x+512,16*x^4+4*x^3-128*x^2-144*x+229,
x^6-4*x^5+7*x^3-x^2-3*x+1,
x^10-36*x^9+540*x^8-4212*x^7+16875*x^6-23679*x^5-43659*x^4+114669*x^3
+96228*x^2-58320*x-46656],
[y-1,y,
(16*x^4+4*x^3-128*x^2-144*x+229)*y^6
+((-88*x^4)-104*x^3+782*x^2+636*x-606)*y^5
+(8*x^5+281*x^4+146*x^3-1806*x^2-1404*x+363)*y^4
+((-30*x^5)-556*x^4+60*x^3+2342*x^2+1944*x+284)*y^3
+(x^6+78*x^5+556*x^4-208*x^3-1898*x^2-1608*x-405)*y^2
+((-2*x^6)-92*x^5-284*x^4+148*x^3+852*x^2+684*x+162)*y+5*x^6+36*x^5
+48*x^4-46*x^3-144*x^2-108*x-27],
[(y-1)*z^4+x*z^3+((-x*y)+x)*z^2+(y-x-1)*z+y]]]
Lazard's method（その１）
1994年（Unpublished manuscript は 1990年）に Daniel Lazard が発表した（https://link.springer.com/chapter/10.1007/978-1-4612-2628-4_29）CAD の構成方法は，一般化された根の重複度を保つ分割によるもので，その projection set（主係数，定数項，判別式，終結式のみ）は，当時知られていた Collins，McCallum，Hong タイプの何れよりも小型でしたが，正当性の証明に不備がありました．
そのため，2001年に発表された，より小型の McCallum-Brown projection がこれまでの標準でしたが，その正しさが保証されているは well-oriented という条件を満たす場合であるため，他の場合には Hong タイプと使い分ける，或いは，別途多項式を追加するといった制約がありました．
しかし，ここ数年，Scott McCallum らによる検証（https://www.researchgate.net/project/Validity-proof-of-Lazards-method-for-CAD-construction）が進み，遂に 2016年7月，Lazard's method の正当性の証明が発表されたのです（https://arxiv.org/abs/1607.00264）．これは非常に大きな進展であり，本年 6 月の MEGA 2017（https://mega2017.inria.fr/files/2017/06/Parusinski.pdf）に続き，9 月の JARCS 2017（http://www.maths.usyd.edu.au/u/laurent/RCSW/）は今回の研究メンバーが主催者に含まれていることもあり，注目しています．
なお，Lazard' method の lifting では，多項式の係数への sample point の代入を拡張し，sample point を根にもつ 1 次式で割り切れる限り割った商に代入したもの，つまり，sample の変数についての multi-index（Lazard valuation）が辞書式順序で最小である 0 でない偏微分係数の 0 でない定数倍（Lazard evaluation）を用います（実装においては，この処理は多項式が消失する場合のみで十分です）．
実行例：d = 0, c = 0 のとき，256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4-4*b^3*c^2 は消失しますが，上記の処理により，Lazard evaluation -4*b^3（実行例の 72*a*b*d-81*a*c^2-2*b^3 に対応）が得られます．
(%i1) (qvpeds ([all],[d,c,a,b,x],0,l,r11,0 ),
g1:qe( bfpcad(ext( '( a*x^4+b*x^2+c*x+d>=0 ) ))) );
Evaluation took 3.6700 seconds (3.6800 elapsed) using 506.869 MB.
(%o1) [[d = root(d,1),c = root(c,1),root(a,1) <= a,
root(72*a*b*d-81*a*c^2-2*b^3,1) <= b,true],
[root(d,1) < d,c < root(c,1),root(a,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,2)
<= b,true],
[root(d,1) < d,c = root(c,1),root(4096*a*d^3+27*c^4,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,1)
<= b,true],
[root(d,1) < d,root(c,1) < c,root(a,1) <= a,
root(256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4
-4*b^3*c^2,2)
<= b,true]]
(%i2) %pp;
Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
(%o2) [[d],[c],[a,256*a*d^3-27*c^4,4096*a*d^3+27*c^4],
[256*a^2*d^3-128*a*b^2*d^2+(144*a*b*c^2+16*b^4)*d-27*a*c^4-4*b^3*c^2],
[a*x^4+b*x^2+c*x+d]]
(%i3) laz_eval([[],[d=0,c=0,a=-1/2]],%pp[4]);
Evaluation took 0.0000 seconds (0.0100 elapsed) using 368.906 KB.
(%o3) [[b],[4*(72*a*b*d-81*a*c^2-2*b^3)]]
(%i4) qex([],F2G(g1) %eq qex([[A,x]], a*x^4+b*x^2+c*x+d>=0 ) );
Evaluation took 30.2000 seconds (35.1300 elapsed) using 4898.136 MB.
(%o4) true
結果の検証
今回の QE ツールにおける入出力（に対応した論理式）の等価性の検証は，いつもお世話になっている qepmax（https://github.com/YasuakiHonda/qepmax）を介して QEPCAD B で行なっています．具体的には，次のように出力のリストを論理結合に変換する関数 F2G（内部で QEPCAD B を繰り返し呼んでいます）で処理したのち，QEPCAD B の出力と等価であるかをチェックしています．
(%i1) (qvpeds ([all],[c,b,a,x],0,h1,r11,0 ),G1:qe( bfpcad(ext( '( x^4+a*x^2+b*x+c>=0 ) ))) )$
Evaluation took 6.6300 seconds (6.7100 elapsed) using 828.758 MB.
(%i2) G1:F2G(G1);
Evaluation took 17.2500 seconds (20.6700 elapsed) using 2096.781 MB.
(%o2) (((a >= 0) %and (c > 0)) %or ((a >= 0) %and (4096*c^3+27*b^4 = 0))
%or ((c > 0)
%and ((-64*a*c^2)+36*b^2*c+16*a^3*c-3*a^2*b^2
>= 0)))
%and (c >= 0)
%and (256*c^3-128*a^2*c^2+144*a*b^2*c+16*a^4*c-27*b^4-4*a^3*b^2 >= 0)
(%i3) qex([],G1 %eq qex([[A,x]],x^4+a*x^2+b*x+c>=0));
Evaluation took 3.2500 seconds (3.6800 elapsed) using 438.299 MB.
(%o3) true
ただし，well-oriented という条件を満たさない入力の場合，QEPCAD B のデフォルトの projection では正当性が保証されないので，Hong's projection operator を併用するコマンドを挟むよう qepmax の qe コマンドの定義（https://github.com/YasuakiHonda/qepmax/blob/master/qepmax.mac）の 214 行目を
if length(varlist)>1 then
( printf(ost, "[ ~a ].~%proj-operator (m", writeLogicalExp(formula,varlist)),
if length(varlist)>2 then printf(ost, ",m"),
for i:1 thru length(varlist)-3 do printf(ost, ",h"),
printf(ost, ")~%finish~%")
)
else printf(ost, "[ ~a ].~%finish~%", writeLogicalExp(formula,varlist)),
のように改変し qex（以前公開した同名の関数とは別物）として使わせて頂いています．
実行例（４）
(%i1) (qvpeds ([],[a,b,c,d],0,h1,r11,0 ),
qe( bfpcad(ext( '( a^3+b^2-1=0 and b^3+c^2-1=0 and c^3+d^2-1=0 and d^3+a^2-1=0 ) ))) );
Evaluation took 14.3200 seconds (16.7700 elapsed) using 2101.559 MB.
(%o1) [[a = root(a,1),b = root(b^2+a^3-1,2),
c = root(c^9-3*c^6+3*c^3+a^4-2*a^2,1),d = root(d^3+a^2-1,1)],
[a = root(a^3+a^2-1,1),
b = root(b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12
+a^4*(6*b^9-18*b^6+18*b^3-2)+33*b^9
+a^2*((-12*b^9)+36*b^6-36*b^3+12)-18*b^6+18*b^3+a^8-4*a^6
-7,1),c = root(c^9-3*c^6+3*c^3+a^4-2*a^2,1),
d = root(d^2+c^3-1,2)],
[a = root(a-1,1),b = root(b^2+a^3-1,1),c = root(c^2+b^3-1,2),
d = root(d^3+a^2-1,1)]]
(%i2) (qvpeds ([],[a,b,c,d],1,h1,r11,0 ),
qe( bfpcad(ext( '( a^3+b^2-1=0 and b^3+c^2-1=0 and c^3+d^2-1=0 and d^3+a^2-1=0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [16,30,1.0b-3,1.0b-16]
equal(b^2+a^3-1,0) and equal(c^2+b^3-1,0) and equal(d^2+c^3-1,0)
and equal(d^3+a^2-1,0)
["and",equal(d^3+a^2-1,0),equal(d^2+c^3-1,0),equal(c^2+b^3-1,0),
equal(b^2+a^3-1,0)]
[[a-1,a,a+1,a^2-2,a^3+a^2-1,a^4-2*a^2-7,a^9-3*a^6+3*a^3-4,
a^25+2*a^24+3*a^23-5*a^22-13*a^21-21*a^20+7*a^19+35*a^18+63*a^17-26*a^16
-115*a^15-204*a^14+31*a^13+266*a^12+501*a^11+115*a^10-271*a^9-657*a^8
-290*a^7+77*a^6+444*a^5+253*a^4+62*a^3-135*a^2-98*a-49,
a^27-9*a^24+36*a^21-81*a^18+108*a^15-81*a^12+9*a^9+54*a^6-54*a^3+33],
[b-1,b,a^3+b^2-1,b^2+b+1,b^6-3*b^3+3,b^9-3*b^6+3*b^3-4,
a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7],
[c-1,c^2+b^3-1,c^2+c+1,c^9-3*c^6+3*c^3+a^4-2*a^2],[d^2+c^3-1,d^3+a^2-1]]
1 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+75*b^9-144*b^6+144*b^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+75*b^9-144*b^6+144*b^3]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots: (-8.660804423214204b-1*%i)-5.000441678406675b-1
(-8.659596311546991b-1*%i)-5.000255856711448b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.222318903917476b-4
2 multi-roots: (-8.659596311546991b-1*%i)-5.000255856711448b-1
(-8.660361378779625b-1*%i)-4.999302464884673b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.222409033757573b-4
3 multi-roots: 8.660599669975173b-1*%i-5.000356595414205b-1
8.659772383882947b-1*%i-5.000121050417724b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 8.601649398563011b-5
4 multi-roots: 8.659772383882947b-1*%i-5.000121050417724b-1
8.660390059677586b-1*%i-4.999522354169911b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 8.602096165627294b-5
5 multi-roots: 2.231315975623363b-5*%i+9.999456693238325b-1
1.000007845795338b0-5.8207805235442b-5*%i
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.017326860553025b-4
6 multi-roots: 1.000007845795338b0-5.8207805235442b-5*%i
3.589464596707269b-5*%i+1.000046484881349b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.017263500284118b-4
1 multi-roots: (-9.199710801626131b-1*%i)-7.71947399037216b-1
(-9.199710804360586b-1*%i)-7.719473929391142b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 6.104229478123359b-9
2 multi-roots: (-9.199710804360586b-1*%i)-7.719473929391142b-1
(-9.199710825477433b-1*%i)-7.719473924291037b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 2.172400328519447b-9
3 multi-roots: 9.199708723835111b-1*%i-7.719494718698244b-1
9.19972984231248b-1*%i-7.71946537008062b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.615703863455374b-6
4 multi-roots: 9.19972984231248b-1*%i-7.71946537008062b-1
9.199693865316476b-1*%i-7.719461755275532b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.61581395716781b-6
5 multi-roots: (-1.128504487190754b0*%i)-4.107493297893198b-1
(-1.128519218977068b0*%i)-4.107484344965202b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.475896599372942b-5
6 multi-roots: (-1.128519218977068b0*%i)-4.107484344965202b-1
(-1.128511078256128b0*%i)-4.107361243173334b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.475845008925132b-5
7 multi-roots: 1.128511588695093b0*%i-4.107446393024692b-1
1.128511596096971b0*%i-4.107446267335344b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.458649779375334b-8
8 multi-roots: 1.128511596096971b0*%i-4.107446267335344b-1
1.128511599631896b0*%i-4.107446225671621b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 5.463913988241733b-9
9 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
10 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
11 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
12 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
13 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
14 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
15 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
16 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
17 multi-roots: 1.182691836749522b0-2.085404757701443b-1*%i
1.182692087088606b0-2.085407026268325b-1*%i
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.378366674990088b-7
18 multi-roots: 1.182692087088606b0-2.085407026268325b-1*%i
1.182692149170476b0-2.085403628805644b-1*%i
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.453718073079048b-7
19 multi-roots: 2.085405167216828b-1*%i+1.182692020651834b0
2.085404528573458b-1*%i+1.182692021812494b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 6.387488296457688b-8
20 multi-roots: 2.085404528573458b-1*%i+1.182692021812494b0
2.085405716985108b-1*%i+1.18269203054428b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.191615147068571b-7
common-roots: 1.346676703722613b0 1.346676703722613b0
dist: 1.665334536937735b-16
1 multi-roots: (-8.660804423214204b-1*%i)-5.000441678406675b-1
(-8.659596311546991b-1*%i)-5.000255856711448b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.222318903917476b-4
2 multi-roots: (-8.659596311546991b-1*%i)-5.000255856711448b-1
(-8.660361378779625b-1*%i)-4.999302464884673b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.222409033757573b-4
3 multi-roots: 8.660599669975173b-1*%i-5.000356595414205b-1
8.659772383882947b-1*%i-5.000121050417724b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 8.601649398563011b-5
4 multi-roots: 8.659772383882947b-1*%i-5.000121050417724b-1
8.660390059677586b-1*%i-4.999522354169911b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 8.602096165627294b-5
5 multi-roots: 2.231315975623363b-5*%i+9.999456693238325b-1
1.000007845795338b0-5.8207805235442b-5*%i
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.017326860553025b-4
6 multi-roots: 1.000007845795338b0-5.8207805235442b-5*%i
3.589464596707269b-5*%i+1.000046484881349b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7]
dist: 1.017263500284118b-4
common-roots: 1.0b0 1.0b0
dist: 0.0b0
common-roots: 7.548776662466926b-1 7.548776662466838b-1
dist: 8.743006318923108b-15
1 multi-roots: 0.0b0 0.0b0 [a^3+b^2-1,b^2]
dist: 0.0b0
1 multi-roots: (-9.199710801626131b-1*%i)-7.71947399037216b-1
(-9.199710804360586b-1*%i)-7.719473929391142b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 6.104229478123359b-9
2 multi-roots: (-9.199710804360586b-1*%i)-7.719473929391142b-1
(-9.199710825477433b-1*%i)-7.719473924291037b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 2.172400328519447b-9
3 multi-roots: 9.199708723835111b-1*%i-7.719494718698244b-1
9.19972984231248b-1*%i-7.71946537008062b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.615703863455374b-6
4 multi-roots: 9.19972984231248b-1*%i-7.71946537008062b-1
9.199693865316476b-1*%i-7.719461755275532b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.61581395716781b-6
5 multi-roots: (-1.128504487190754b0*%i)-4.107493297893198b-1
(-1.128519218977068b0*%i)-4.107484344965202b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.475896599372942b-5
6 multi-roots: (-1.128519218977068b0*%i)-4.107484344965202b-1
(-1.128511078256128b0*%i)-4.107361243173334b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.475845008925132b-5
7 multi-roots: 1.128511588695093b0*%i-4.107446393024692b-1
1.128511596096971b0*%i-4.107446267335344b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.458649779375334b-8
8 multi-roots: 1.128511596096971b0*%i-4.107446267335344b-1
1.128511599631896b0*%i-4.107446225671621b-1
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 5.463913988241733b-9
9 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
10 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
11 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
12 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
13 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
14 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
15 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
16 multi-roots: 0.0b0 0.0b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 0.0b0
17 multi-roots: 1.182691836749522b0-2.085404757701443b-1*%i
1.182692087088606b0-2.085407026268325b-1*%i
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.378366674990088b-7
18 multi-roots: 1.182692087088606b0-2.085407026268325b-1*%i
1.182692149170476b0-2.085403628805644b-1*%i
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 3.453718073079048b-7
19 multi-roots: 2.085405167216828b-1*%i+1.182692020651834b0
2.085404528573458b-1*%i+1.182692021812494b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 6.387488296457688b-8
20 multi-roots: 2.085404528573458b-1*%i+1.182692021812494b0
2.085405716985108b-1*%i+1.18269203054428b0
[a^8-4*a^6+(6*b^9-18*b^6+18*b^3-2)*a^4+((-12*b^9)+36*b^6-36*b^3+12)*a^2+b^27
-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+33*b^9-18*b^6+18*b^3-7,
b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12+27*b^9]
dist: 1.191615147068571b-7
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots: 9.802492832500017b-24*%i-3.332445010132976b-6
1.666222505066488b-6-2.885982035489848b-6*%i
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3+1.110223024625157b-16]
dist: 5.771964070979697b-6
2 multi-roots: 1.666222505066488b-6-2.885982035489848b-6*%i
2.885982035489848b-6*%i+1.666222505066488b-6
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3+1.110223024625157b-16]
dist: 5.771964070979697b-6
1 multi-roots: (-8.660255628474419b-1*%i)-5.00000111173379b-1
(-8.660252247378837b-1*%i)-5.000000835080218b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1.0b0]
dist: 3.392395102543312b-7
2 multi-roots: (-8.660252247378837b-1*%i)-5.000000835080218b-1
(-8.660254237679908b-1*%i)-4.999998053185997b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1.0b0]
dist: 3.420560452027647b-7
3 multi-roots: 8.660254036658418b-1*%i-5.000000000446686b-1
8.660254038379494b-1*%i-4.999999999906503b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1.0b0]
dist: 1.803856285841607b-10
4 multi-roots: 8.660254038379494b-1*%i-4.999999999906503b-1
8.660254038495248b-1*%i-4.999999999646812b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1.0b0]
dist: 2.843207457168711b-11
5 multi-roots: 1.632012483901229b-10*%i+9.999999990929849b-1
1.000000000452425b0-8.155697400955311b-11*%i
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1.0b0]
dist: 1.381297987296357b-9
6 multi-roots: 1.000000000452425b0-8.155697400955311b-11*%i
1.000000000454591b0-8.164377541315077b-11*%i
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1.0b0]
dist: 2.167145783155763b-12
1 multi-roots: 0.0b0 0.0b0 [c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [c^2+b^3-1,c^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 7.548776662466963b-1 7.548776662466864b-1
dist: 9.867107131356079b-15
1 multi-roots: (-8.660255628474419b-1*%i)-5.00000111173379b-1
(-8.660252247378837b-1*%i)-5.000000835080218b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1]
dist: 3.392395102543312b-7
2 multi-roots: (-8.660252247378837b-1*%i)-5.000000835080218b-1
(-8.660254237679908b-1*%i)-4.999998053185997b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1]
dist: 3.420560452027647b-7
3 multi-roots: 8.660254036658418b-1*%i-5.000000000446686b-1
8.660254038379494b-1*%i-4.999999999906503b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1]
dist: 1.803856285841607b-10
4 multi-roots: 8.660254038379494b-1*%i-4.999999999906503b-1
8.660254038495248b-1*%i-4.999999999646812b-1
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1]
dist: 2.843207457168711b-11
5 multi-roots: 1.632012483901229b-10*%i+9.999999990929849b-1
1.000000000452425b0-8.155697400955311b-11*%i
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1]
dist: 1.381297987296357b-9
6 multi-roots: 1.000000000452425b0-8.155697400955311b-11*%i
1.000000000454591b0-8.164377541315077b-11*%i
[c^9-3*c^6+3*c^3+a^4-2*a^2,c^9-3*c^6+3*c^3-1]
dist: 2.167145783155763b-12
common-roots: 1.0b0 1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [d^3+a^2-1,d^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [d^3+a^2-1,d^3]
dist: 0.0b0
common-roots: 1.0b0 1.0b0
dist: 0.0b0
common-roots: 7.548776662466929b-1 7.548776662466925b-1
dist: 4.440892098500626b-16
1 multi-roots: 0.0b0 0.0b0 [d^3+a^2-1,d^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [d^3+a^2-1,d^3]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [d^2+c^3-1,d^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [d^3+a^2-1,d^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [d^3+a^2-1,d^3]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
[T,F]: [3,440]
Evaluation took 14.8600 seconds (17.4500 elapsed) using 2063.632 MB.
(%o2) [[a = root(a,1),b = root(b^2+a^3-1,2),
c = root(c^9-3*c^6+3*c^3+a^4-2*a^2,1),d = root(d^3+a^2-1,1)],
[a = root(a^3+a^2-1,1),
b = root(b^27-9*b^24+36*b^21-81*b^18+108*b^15-81*b^12
+a^4*(6*b^9-18*b^6+18*b^3-2)+33*b^9
+a^2*((-12*b^9)+36*b^6-36*b^3+12)-18*b^6+18*b^3+a^8-4*a^6
-7,1),c = root(c^9-3*c^6+3*c^3+a^4-2*a^2,1),
d = root(d^2+c^3-1,2)],
[a = root(a-1,1),b = root(b^2+a^3-1,1),c = root(c^2+b^3-1,2),
d = root(d^3+a^2-1,1)]]
実行例（３）
CGS-EQ の深作亮也先生（東京理科大）のサイト
http://www.rs.tus.ac.jp/fukasaku/software/CGSQE-20160509/benchmark/computation-time/
http://www.rs.tus.ac.jp/fukasaku/software/CGSQE-20160509/benchmark/input/04/log/
出典
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.642.8545&rep=rep1&type=pdf Example 9
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([ex,ex],[x,y,z],0,0,r11,0 ),
qe( bfpcad(ext( '( x^2+y^2*z+z^3=0 and 3*x^2+3*y^2+z^2-1=0 and x^2+z^2-y^3*(y-1)^3<0 ) ))) );
Evaluation took 34.9400 seconds (34.9500 elapsed) using 4777.359 MB.
(%o1) [[root(387420489*x^36+473513931*x^34+1615049199*x^32-5422961745*x^30
+2179233963*x^28-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8-446456947*x^6
+43657673*x^4-1631864*x^2-40328,1)
< x
and x < root(387420489*x^36+473513931*x^34+1615049199*x^32
-5422961745*x^30+2179233963*x^28
-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8
-446456947*x^6+43657673*x^4-1631864*x^2
-40328,2),true,true]]
(%i2) (qvpeds ([ex,ex],[x,y,z],1,0,r11,0 ),
qe( bfpcad(ext( '( x^2+y^2*z+z^3=0 and 3*x^2+3*y^2+z^2-1=0 and x^2+z^2-y^3*(y-1)^3<0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [16,30,1.0b-3,1.0b-16]
equal(z^3+y^2*z+x^2,0) and equal(z^2+3*y^2+3*x^2-1,0)
and z^2-y^6+3*y^5-3*y^4+y^3+x^2 < 0
["and",z^2-y^6+3*y^5-3*y^4+y^3+x^2 < 0,equal(z^2+3*y^2+3*x^2-1,0),
equal(z^3+y^2*z+x^2,0)]
[[z^2+3*y^2+3*x^2-1],[z^2-y^6+3*y^5-3*y^4+y^3+x^2],[z^3+y^2*z+x^2]]
[[3*y^2+3*x^2-1],[y^6-3*y^5+3*y^4-y^3-x^2],
[y^6-3*y^5+3*y^4-y^3+3*y^2+2*x^2-1],[4*y^6+27*x^4],
[12*y^6+48*x^2*y^4-16*y^4+63*x^4*y^2-42*x^2*y^2+7*y^2+27*x^6-26*x^4+9*x^2-1],
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13-3*x^2*y^12+114*y^12
+18*x^2*y^11-76*y^11-45*x^2*y^10+40*y^10+60*x^2*y^9-16*y^9-49*x^2*y^8
+5*y^8+30*x^2*y^7-y^7+3*x^4*y^6-15*x^2*y^6-9*x^4*y^5+4*x^2*y^5+9*x^4*y^4
-x^2*y^4-3*x^4*y^3+2*x^4*y^2-x^6-x^4]]
[[x,x^2-2,2*x^2-1,3*x^2-1,27*x^6-155*x^4+9*x^2-1,27*x^6-26*x^4+9*x^2-1,
54*x^6-297*x^4+18*x^2-2,108*x^6-837*x^4+36*x^2-4,
729*x^12+729*x^10+1215*x^8-6129*x^6+4185*x^4-504*x^2-8,
387420489*x^36+473513931*x^34+926301663*x^32-4590725139*x^30+3855398877*x^28
-5886383967*x^26+14051402938*x^24-13565732679*x^22
+9151246767*x^20-15472018962*x^18+21618673717*x^16
-15614937665*x^14+6066412333*x^12-1269760998*x^10
+127675066*x^8-3130295*x^6-196075*x^4-2296*x^2-8,
387420489*x^36+473513931*x^34+1615049199*x^32-5422961745*x^30
+2179233963*x^28-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20-126478283472*x^18
+164389796305*x^16-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8-446456947*x^6+43657673*x^4
-1631864*x^2-40328,
387420489*x^36+1162261467*x^34+4649045868*x^32-9125904852*x^30
-4864279473*x^28-75245668308*x^26+228853906389*x^24
-233182493334*x^22+526458209184*x^20-1632367306233*x^18
+2577322981359*x^16-2213846330418*x^14+1103473933785*x^12
-328088633355*x^10+61540990272*x^8-7555532337*x^6
+598204197*x^4-19509336*x^2-512072],
[3*y^2+3*x^2-1,y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3+3*y^2+2*x^2-1,
4*y^6+27*x^4,
12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4],
[z^2+3*y^2+3*x^2-1,z^2-y^6+3*y^5-3*y^4+y^3+x^2,z^3+y^2*z+x^2]]
1 multi-roots: 0.0b0 0.0b0 [3*y^2+3*x^2-1,3*y^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-5.72536737891363b0*y^4+8.963235209558525b-1*y^2]
dist: 0.0b0
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
common-roots: -3.731302026563126b-1 -3.731302026563127b-1
dist: 4.163336342344337b-17
common-roots: -4.492182869390443b-1 -4.492182869390443b-1
dist: 6.938893903907228b-17
common-roots: -4.293461173741275b-1 -4.293461173741276b-1
dist: 4.163336342344337b-17
common-roots: -4.293461173741276b-1 -4.293461173741276b-1
dist: 2.081668171172169b-17
common-roots: 1.242338414099984b0 1.241578377687144b0
dist: 7.600364128400594b-4
1 multi-roots: 0.0b0 0.0b0 [y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 0.0b0
3 multi-roots: 9.999999999999998b-1-1.908195823574488b-16*%i
1.006139616066548b-16*%i+1.0b0 [y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 4.425687238657765b-16
4 multi-roots: 1.006139616066548b-16*%i+1.0b0 9.020562075079397b-17*%i+1.0b0
[y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 1.040834085586084b-17
1 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
4 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
5 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
1 multi-roots: (-3.118675378203367b-11*%i)-7.071067811740978b-1
3.11867695771263b-11*%i-7.071067811989973b-1
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-16*y^4+7*y^2-1]
dist: 6.715982037737266b-11
2 multi-roots: 7.071067811770959b-1-1.273326148703733b-11*%i
1.273326308308289b-11*%i+7.071067811959992b-1
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-16*y^4+7*y^2-1]
dist: 3.171562834653794b-11
1 multi-roots: (-6.025654200204613b-1*%i)-1.891279438990948b-2
(-6.025654199767368b-1*%i)-1.891279438040217b-2
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 4.474615144603693b-11
2 multi-roots: 6.025654209207471b-1*%i-1.891280116685924b-2
6.025654190764511b-1*%i-1.89127876034522b-2
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 1.368822261249517b-8
3 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
4 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
5 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
6 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
7 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
8 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
9 multi-roots: 2.717041990286581b-6*%i+9.99993414966278b-1
1.00000093787557b0-7.060909170695758b-6*%i
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 1.233703745313714b-5
10 multi-roots: 1.00000093787557b0-7.060909170695758b-6*%i
4.343867253830464b-6*%i+1.000005647158198b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 1.233881145655348b-5
11 multi-roots: 1.518912661150419b0-6.666097968941921b-1*%i
1.518912927619844b0-6.666098929698847b-1*%i
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 2.832604685158154b-7
12 multi-roots: 6.666098479459857b-1*%i+1.51891279382378b0
6.666098419180176b-1*%i+1.518912794946535b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 6.131637614094183b-9
common-roots: 1.000000000000015b0 1.0b0
dist: 1.515454428613339b-14
common-roots: 5.773502691896258b-1 5.773502691896257b-1
dist: 5.551115123125783b-17
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -5.773502691896258b-1 -5.773502691896258b-1
dist: 5.551115123125783b-17
common-roots: 1.242338414100015b0 1.241578377687144b0
dist: 7.600364128707848b-4
common-roots: -4.293461173741275b-1 -4.293461173741275b-1
dist: 1.387778780781446b-17
common-roots: -4.293461173741275b-1 -4.293461173741276b-1
dist: 3.469446951953614b-17
common-roots: -4.492182869390442b-1 -4.492182869390443b-1
dist: 9.020562075079397b-17
common-roots: -3.731302026563125b-1 -3.731302026563127b-1
dist: 2.012279232133096b-16
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
1 multi-roots: 0.0b0 0.0b0
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-5.72536737891363b0*y^4+8.963235209558528b-1*y^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [3*y^2+3*x^2-1,3*y^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 1.9369471621906b-1 1.9369471621906b-1
dist: 0.0b0
common-roots: -1.9369471621906b-1 -1.9369471621906b-1
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.704199308450861b-1 2.704199308450861b-1
dist: 0.0b0
common-roots: -2.704199308450861b-1 -2.704199308450861b-1
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.755098124531884b-1 2.755098124531884b-1
dist: 0.0b0
common-roots: -2.755098124531884b-1 -2.755098124531884b-1
dist: 0.0b0
common-roots: -4.190704888041301b-1 -4.190704888041305b-1
dist: 3.747002708109903b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.799706718183064b-1 -3.799706718183065b-1
dist: 4.85722573273506b-17
common-roots: 2.804840673490828b-1 2.804840673490828b-1
dist: 0.0b0
common-roots: -2.804840673490828b-1 -2.804840673490828b-1
dist: 0.0b0
common-roots: -4.152590546912595b-1 -4.152590546912597b-1
dist: 1.179611963664229b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.655627214498231b-1 -3.655627214498232b-1
dist: 5.551115123125783b-17
common-roots: 3.184417034569177b-1 3.184417034569177b-1
dist: 0.0b0
common-roots: -3.184417034569177b-1 -3.184417034569177b-1
dist: 0.0b0
common-roots: -3.831359850498015b-1 -3.831359850498017b-1
dist: 1.52655665885959b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 3.509771861657056b-1 3.509771861657056b-1
dist: 0.0b0
common-roots: -3.509771861657056b-1 -3.509771861657056b-1
dist: 0.0b0
common-roots: -3.509771861657056b-1 -3.509771861657057b-1
dist: 1.110223024625157b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -8.666384296627297b-2 -8.666384296627914b-2
dist: 6.168676680573526b-15
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -8.666384296627296b-2 -8.666384296627922b-2
dist: 6.257147577848343b-15
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 3.509771861657057b-1 3.509771861657057b-1
dist: 0.0b0
common-roots: -3.509771861657057b-1 -3.509771861657057b-1
dist: 0.0b0
common-roots: -3.509771861657057b-1 -3.509771861657057b-1
dist: 2.081668171172169b-17
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.655627214498231b-1 -3.655627214498232b-1
dist: 5.551115123125783b-17
common-roots: 3.184417034569177b-1 3.184417034569177b-1
dist: 0.0b0
common-roots: -3.184417034569177b-1 -3.184417034569177b-1
dist: 0.0b0
common-roots: -3.831359850498015b-1 -3.831359850498017b-1
dist: 1.52655665885959b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.799706718183065b-1 -3.799706718183065b-1
dist: 2.081668171172169b-17
common-roots: 2.80484067349083b-1 2.80484067349083b-1
dist: 0.0b0
common-roots: -2.80484067349083b-1 -2.80484067349083b-1
dist: 0.0b0
common-roots: -4.152590546912595b-1 -4.152590546912595b-1
dist: 6.245004513516506b-17
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.755098124531883b-1 2.755098124531883b-1
dist: 0.0b0
common-roots: -2.755098124531883b-1 -2.755098124531883b-1
dist: 0.0b0
common-roots: -4.190704888041303b-1 -4.190704888041304b-1
dist: 1.110223024625157b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.704199308450858b-1 2.704199308450858b-1
dist: 0.0b0
common-roots: -2.704199308450858b-1 -2.704199308450858b-1
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 1.936947162190598b-1 1.936947162190598b-1
dist: 0.0b0
common-roots: -1.936947162190598b-1 -1.936947162190598b-1
dist: 0.0b0
[T,F]: [3,1064]
Evaluation took 32.3300 seconds (32.4100 elapsed) using 4531.459 MB.
(%o2) [[root(387420489*x^36+473513931*x^34+1615049199*x^32-5422961745*x^30
+2179233963*x^28-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8-446456947*x^6
+43657673*x^4-1631864*x^2-40328,1)
< x
and x < root(387420489*x^36+473513931*x^34+1615049199*x^32
-5422961745*x^30+2179233963*x^28
-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8
-446456947*x^6+43657673*x^4-1631864*x^2
-40328,2),true,true]]
実行例（２）
SyNRAC の岩根秀直さん（富士通研究所）のサイト
https://github.com/hiwane/qe_problems/blob/master/problems/exam/manual-fof/tsukuba2010-Ri-1-m.mpl
他の QE ツールの出力との比較
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2019-13.pdf Appendix
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([ex],[a,b,x],0,0,r11,0 ),
qe( bfpcad(ext( '( -1 <= x and x <= 3 and b = x^3/3 - a*x^2/2 and a > 0 ) ))) );
Evaluation took 1.7400 seconds (1.7400 elapsed) using 188.173 MB.
(%o1) [[root(a,1) < a and a < root(a-2,1),
root(6*b+3*a+2,1) <= b and b <= root(2*b+9*a-18,1),true],
[a = root(a-2,1),root(6*b+a^3,1) <= b and b <= root(2*b+9*a-18,1),
true],
[root(a-2,1) < a and a <= root(a-3,1),
root(6*b+a^3,1) <= b and b <= root(b,1),true],
[root(a-3,1) < a,root(2*b+9*a-18,1) <= b and b <= root(b,1),true]]
(%i2) (qvpeds ([ex],[a,b,x],1,0,r11,0 ),
qe( bfpcad(ext( '( -1 <= x and x <= 3 and b = x^3/3 - a*x^2/2 and a > 0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [16,30,1.0b-3,1.0b-16]
x+1 >= 0 and x-3 <= 0 and equal((2*x^3-3*a*x^2-6*b)/6,0) and a > 0
["and",equal((2*x^3-3*a*x^2-6*b)/6,0),x-3 <= 0,x+1 >= 0,a > 0]
[[x-3],[x+1],[2*x^3-3*a*x^2-6*b]]
[[b],[2*b+9*a-18],[6*b+3*a+2],[6*b+a^3]]
[[a-3,a-2,a,a+1,a+6,3*a-7,3*a+2],[b,2*b+9*a-18,6*b+3*a+2,6*b+a^3],
[x-3,x+1,2*x^3-3*a*x^2-6*b]]
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -1.333333333333333b0 -1.333333333333333b0
dist: 0.0b0
common-roots: -1.5b0 -1.5b0
dist: 0.0b0
common-roots: -4.5b0 -4.5b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 9.999999999999999b-1-3.469517903370656b-17*%i
3.469446951953614b-17*%i+1.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-3.0b0*x^2+1.0b0]
dist: 1.309231643767475b-16
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-3.0b0*x^2]
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 6.938893903907228b-17*%i+2.0b0 2.0b0-6.939035806741312b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-6*x^2+8]
dist: 2.61846328753495b-16
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-6*x^2]
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 2.166666666666667b0-4.163490115220211b-17*%i
4.163336342344337b-17*%i+2.166666666666667b0
[2*x^3-3*a*x^2-6*b,2*x^3-6.5b0*x^2+1.01712962962963b1]
dist: 1.861907491918325b-16
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-6.5b0*x^2]
dist: 0.0b0
1 multi-roots: 2.333333333333333b0-5.551280766043446b-17*%i
5.551115123125783b-17*%i+2.333333333333334b0
[2*x^3-3*a*x^2-6*b,2*x^3-7*x^2+1.27037037037037b1]
dist: 1.171009175116753b-15
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-7*x^2]
dist: 0.0b0
1 multi-roots: 1.387778780781446b-17*%i+2.666666666666667b0
2.666666666666667b0-1.387968122423659b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-8.0b0*x^2+1.896296296296296b1]
dist: 1.144396291987333b-16
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-8.0b0*x^2]
dist: 0.0b0
1 multi-roots: 3.0b0-4.163336342344337b-17*%i 4.163123343336605b-17*%i+3.0b0
[2*x^3-3*a*x^2-6*b,2*x^3-9*x^2+27]
dist: 2.371429941323172b-16
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-9*x^2]
dist: 0.0b0
1 multi-roots: 5.550866638069772b-17*%i+3.5b0 3.5b0-5.551115123125783b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-1.05b1*x^2+4.2875b1]
dist: 5.661043993863231b-16
common-roots: 3.0b0 3.0b0
dist: 5.551115123125783b-17
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-1.05b1*x^2]
dist: 0.0b0
[T,F]: [39,253]
Evaluation took 2.5500 seconds (2.5600 elapsed) using 169.662 MB.
(%o2) [[root(a,1) < a and a < root(a-2,1),
root(6*b+3*a+2,1) <= b and b <= root(2*b+9*a-18,1),true],
[a = root(a-2,1),root(6*b+a^3,1) <= b and b <= root(2*b+9*a-18,1),
true],
[root(a-2,1) < a and a <= root(a-3,1),
root(6*b+a^3,1) <= b and b <= root(b,1),true],
[root(a-3,1) < a,root(2*b+9*a-18,1) <= b and b <= root(b,1),true]]
projection set に対する数値解の誤差（その２）
．．．とは言え，sep の評価なしでは，いくら R を小さくしても，前回の冒頭で述べたように，近接根に対応した数値解をカウントしてしまう可能性を排除出来ないかのように見えます．
しかし，我々が扱うのは一般の系ではなく，CAD の projection set であり，特に pscs を含む Collins-Hong タイプの projection（定義は，例えば，D.Wilson. Advances in Cylindrical Algebraic Decomposition の 2.3.1 Collins’Algorithm，2.4.1 Alternative Projection Operators をご参照ください）を利用すれば，sep の情報なしに，重根に対応した数値解全体のみを正確にカウントすることが出来ます．以下，そのことをお話しましょう．
まず，pscs（principal subresultant coefficient sequence）とは，多項式 f(x), g(x) に対して，それらの Sylvester 行列
syl(f1,f2,v):=block([d1:hipow(f1,v),d2:hipow(f2,v),c1,c2,z0,i,k],
c1:makelist(coeff(f1,v,i),i,d1,0,-1),
c2:makelist(coeff(f2,v,i),i,d2,0,-1),z0:makelist(0,d1+d2),
[makelist(firstn(append(firstn(z0,k-1),c1,z0),d1+d2),k,d2),
makelist(firstn(append(firstn(z0,k-1),c2,z0),d1+d2),k,d1)])$
の一部を除いた行列の行列式の列（普通の resultant はその初項です）
pscs(f01,f02,v):=block(
[uratmx:ratmx,s1,s2,d,d1,d2,f1:expand(f01),f2:expand(f02),j],
ratmx:on,[s1,s2]:syl(f1,f2,v),d1:hipow(f1,v),d2:hipow(f2,v),
d:makelist(determinant(apply(matrix,
append(
map(lambda([e],firstn(e,d1+d2-2*j)),
firstn(s1,d2-j)),
map(lambda([e],firstn(e,d1+d2-2*j)),
firstn(s2,d1-j))))),j,0,
min(d1,d2)-1),ratmx:uratmx,d)$
のことで
( pscs の初項から続く 0 の個数 ) = ( gcd( f(x), g(x) ) の次数 )
という性質（Polynomial greatest common divisor - Wikipedia）を持ちます．例えば，上の（素朴な）コードを読み込んで，実行すると
(%i24) pscs( (x-1) * (x-2) , (x-3) * (x-4), x ); (%o24) [12,-4] (%i25) pscs( (x-1) * (x-2)^3 * (x-3)^2 , (x-2)^2 * (x-3)^3 * (x-4), x ); (%o25) [0,0,0,0,12,-4] (%i26) factor(pscs( (x-1) * (x-a)^3 * (x-b)^2 , (x-a)^2 * (x-b)^3 * (x-4), x )); (%o26) [0,0,0,0,-3*(a-4)*(b-1)*(b-a),-(b-a+3)]
といった具合になります．
さて，上の性質を g(x) = f'(x) の場合に用いると
gcd( f(x), f'(x) ) の次数
の値が pscs から特定できる訳ですが，この値が f(x) の
(重根の個数) - (相異なる重根の個数)
であることは，積の微分公式により簡単に判ります．例で確認してみると
(%i31) pscs( f : (x-1) * (x-2)^3 * (x-3)^5 , diff(f,x) , x ); (%o31) [0,0,0,0,0,0,-60,-38]
のように 8 - 2 = 6 となっています．
ところが，この差は前回の冒頭で述べた方法における，重根に対応する数値解の検出カウントに他なりません．例えば，上の重根 2, 2, 2, 3, 3, 3, 3, 3 に対応した数値解 s1, s2, s3, t1, t2, t3, t4, t5 は
2*R >= | s1 - s2 |, 2*R >= | s2 - s3 |,
2*R >= | t1 - t2 |, 2*R >= | t2 - t3 |, 2*R >= | t3 - t4 |, 2*R >= | t4 - t5 |
として，合計 (3-1) + (5-1) = 6 回カウントされ，もし，検出カウントがこれを超えるなら，そこには近接根に対応する数値解のカウントが含まれているので，R を小さくして retry すれば，有限回で近接根に対応する数値解を排除でき，同時に，カウントの長さを見れば，各重根の重複度（各クラスタに属する数値解の個数）も判る，と言う仕組みです．
