GP/PARI CALCULATOR Version 2.14.0 (development git-b0a4238356)
amd64 running linux (x86-64/GMP-6.2.1 kernel) 64-bit version
compiled: Feb 21 2021, gcc version 9.3.0 (Ubuntu 9.3.0-17ubuntu1~20.04)
threading engine: single
(readline v8.0 enabled, extended help enabled)
Copyright (C) 2000-2021 The PARI Group
PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY
WARRANTY WHATSOEVER.
Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.
parisizemax = 10000003072, primelimit = 500000
? tst11Sv2s01([all],[a,b,c,x],id,"a*x^2+b*x+c<=0",2);Ans();pp
[x,1]
[c,2]
[b,1]
[a,1]
var.:a b c x *** connecting adjacent 6/135 cells.
1[a < [a,1],true,c <= [4*c*a-b^2,1],true]
2[a = [a,1],b = [b,1],c <= [c,1],true]
time = 16 ms.
%1 = [[a],[b],[4*c*a-b^2,c],[a*x^2+b*x+c]] /* Lazard's projection */
? tst11Sv2s01([all],[a,b,c,x],id,"a*x^2+b*x+c<=0",2*7);Ans();pp
*** using the sum of squares projection.
[x,1]
[c,2]
[b,1]
[a,1]
var.:a b c x *** connecting adjacent 2/41 cells.
1[a < [a,1],true,c <= [-4*c*a+b^2,1],true]
2[a = [a,1],b = [b,1],c <= [c,1],true]
time = 6 ms.
%2 = [[a],[a^2+b^2],[-4*c*a+b^2,a^2+(b^2+c^2)],[a*x^2+b*x+c]] /* Sum Of Squares projection */
? tst11Sv2s01([ex,ex,ex,ex],[e,v,a,b,c,d],(f1,f2,f3)->f1*f2*f3,"v==(a-1)*(c-1)+b*d,a^2+e*b^2==1,c^2+e*d^2==1",2);Ans();pp
[d,2]
[c,4]
[b,5]
[a,8]
[v,11]
[e,11]
var.:e v a b c d *** connecting adjacent 524/191440 cells.
1[e <= [e,1],true,true,true,true,true]
2[[e,1] < e <= [2*e-1,1],[(e^2+e)*v+1,1] <= v <= [(e^2-e)*v+1,1],true,true,true,true]
3[[2*e-1,1] < e,[(e^2+e)*v+1,1] <= v <= [v-4,1],true,true,true,true]
time = 1min, 55,679 ms.
%3 = [[3*e+1,3*e-1,7*e^4+10*e^2-1,2*e+1,2*e-1,8*e^2-1,3*e^2-1,2*e^2-1,e+1,e-1,e],[(4*e^4-4*e^2+1)*v^2+(8*e^2-4)*v+8,(e^2+e)*v+1,(e^2-e)*v+1,e^2*v^2-2*e^2*v-1,(e^4-2*e^2+1)*v^2+(4*e^2-4)*v+8,(e^2+1)*v-4,(e^2-1)*v+4,v-4,(e^2-1)*v+2,v-2,v],[2*a^2+(2*v-4)*a+(v^2-2*v+2),a^2+2*e^2*v*a+(e^2*v^2-2*e^2*v-1),(e^2+1)*a^2+(2*e^2*v-2*e^2)*a+(e^2*v^2-2*e^2*v+(e^2-1)),(e^2-1)*a+(-e^2-1),2*a+(v-2),a+1,a+(v-1),a-1],[-2*e*v*a+(b^2+(-e*v^2+2*e*v)),-e*a^2+(-2*e*v+2*e)*a+(b^2+(-e*v^2+2*e*v-e)),e*a^2-2*e*a+(b^2+e),a^2+(e*b^2-1),b],[c+1,c-1,(e*c^2-2*e*c+e)*a^2+(-2*e*c^2+(-2*e*v+4*e)*c+(2*e*v-2*e))*a+((c^2-1)*b^2+(e*c^2+(2*e*v-2*e)*c+(e*v^2-2*e*v+e))),(c-1)*a+(-c+(-v+1))],[c^2+(e*d^2-1),(c-1)*a+(d*b+(-c+(-v+1)))]] /* Lazard's projection */
? tst11Sv2s01([ex,ex,ex,ex],[e,v,a,b,c,d],(f1,f2,f3)->f1*f2*f3,"v==(a-1)*(c-1)+b*d,a^2+e*b^2==1,c^2+e*d^2==1",2*7);Ans();pp
*** using the sum of squares projection.
[d,2]
[c,4]
[b,5]
[a,8]
[v,11]
[e,11]
var.:e v a b c d *** connecting adjacent 524/143294 cells.
1[e <= [e,1],true,true,true,true,true]
2[[e,1] < e <= [2*e-1,1],[(e^2+e)*v+1,1] <= v <= [(e^2-e)*v+1,1],true,true,true,true]
3[[2*e-1,1] < e,[(e^2+e)*v+1,1] <= v <= [v-4,1],true,true,true,true]
time = 1min, 32,646 ms.
%4 = [[3*e+1,3*e-1,7*e^4+10*e^2-1,2*e+1,2*e-1,8*e^2-1,3*e^2-1,2*e^2-1,e+1,e-1,e],[(4*e^4-4*e^2+1)*v^2+(8*e^2-4)*v+8,(e^2+e)*v+1,(e^2-e)*v+1,(e^4-2*e^2+1)*v^2+(4*e^2-4)*v+8,e^2*v^2-2*e^2*v-1,(e^2-1)*v+4,(e^2+1)*v-4,v-4,(e^2-1)*v+2,v-2,v],[(e^2+4)*a^4+((2*e^2+8)*v+(-4*e^2-16))*a^3+((e^2+8)*v^2+(-6*e^2-24)*v+(6*e^2+24))*a^2+(4*v^3+(-2*e^2-16)*v^2+(6*e^2+24)*v+(-4*e^2-16))*a+(v^4-4*v^3+(e^2+8)*v^2+(-2*e^2-8)*v+(e^2+4)),(e^2-1)*a+(-e^2-1),a^2+2*e^2*v*a+(e^2*v^2-2*e^2*v-1),(e^4+3*e^2+1)*a^4+((4*e^4+6*e^2)*v+(-4*e^4-8*e^2))*a^3+((6*e^4+3*e^2)*v^2+(-12*e^4-10*e^2)*v+(6*e^4+6*e^2-2))*a^2+(4*e^4*v^3+(-12*e^4-2*e^2)*v^2+(12*e^4+2*e^2)*v-4*e^4)*a+(e^4*v^4-4*e^4*v^3+(6*e^4-e^2)*v^2+(-4*e^4+2*e^2)*v+(e^4-e^2+1)),2*a+(v-2),a+1,a-1,a+(v-1)],[e*a^2-2*e*a+(b^2+e),2*e*v*a+(-b^2+(e*v^2-2*e*v)),(e^2*b^2+e^4)*a^6+(-6*e^2*b^2-6*e^4)*a^5+((e^2+2*e)*b^4+(e^4+2*e^3+15*e^2+1)*b^2+(15*e^4+2*e^2))*a^4+((-4*e^2-8*e)*b^4+(2*v+(-4*e^4-8*e^3-20*e^2-4))*b^2+(6*e^2*v+(-20*e^4-8*e^2)))*a^3+((2*e+1)*b^6+(2*e^3+7*e^2+12*e+1)*b^4+(v^2-6*v+(6*e^4+12*e^3+16*e^2-2*e+6))*b^2+(7*e^2*v^2-18*e^2*v+(15*e^4+12*e^2)))*a^2+((-4*e-2)*b^6+(2*v+(-4*e^3-6*e^2-8*e-2))*b^4+(-2*v^2+(2*e^2-4*e+6)*v+(-4*e^4-8*e^3-8*e^2+4*e-4))*b^2+(4*e^2*v^3-14*e^2*v^2+18*e^2*v+(-6*e^4-8*e^2)))*a+(b^8+(e^2+2*e+1)*b^6+(v^2-2*v+(2*e^3+2*e^2+2*e+2))*b^4+((e^2-2*e+1)*v^2+(-2*e^2+4*e-2)*v+(e^4+2*e^3+2*e^2-2*e+1))*b^2+(e^2*v^4-4*e^2*v^3+7*e^2*v^2-6*e^2*v+(e^4+2*e^2))),a^2+(e*b^2-1),b],[c+1,c-1,(e*c^2-2*e*c+e)*a^2+(-2*e*c^2+(-2*e*v+4*e)*c+(2*e*v-2*e))*a+((c^2-1)*b^2+(e*c^2+(2*e*v-2*e)*c+(e*v^2-2*e*v+e))),(c^2-2*c+1)*a^2+(-2*c^2+(-2*v+4)*c+(2*v-2))*a+(b^2+(c^2+(2*v-2)*c+(v^2-2*v+1)))],[c^2+(e*d^2-1),(c-1)*a+(d*b+(-c+(-v+1)))]] /* Sum Of Squares projection */
