本日のC.A.D.

? tst12([],[a,b,c,x,y],(f1,f2)->f1*f2,"y>=0,y+(a-b)*x^2+(2*a-3*b+c)*a*x+c-b<=0");pp
 *** using Lazard's method (MPP17).
[y,2]
[x,1]
[c,2]
[b,2]
[a,4]
time = 134 ms.
11 57(6,2) 315(36,26) 1219(148,0) 1592(0,946) 
 *** combined adjacent 786 cells.
time = 415 ms.
%1 = [[a,2*a^2-1,3*a-2,3*a+2],[a-b,4*a^4-12*b*a^3+9*b^2*a^2+4*b*a-4*b^2],[b-c,4*a^4+(-12*b+4*c)*a^3+(9*b^2-6*c*b+c^2)*a^2+(4*b-4*c)*a+(-4*b^2+4*c*b)],[(a-b)*x^2+(2*a^2+(-3*b+c)*a)*x+(-b+c)],[y,(a-b)*x^2+(2*a^2+(-3*b+c)*a)*x+(y+(-b+c))]]

? tst12([],[a,b,c,x,y],(f1,f2)->f1*f2,"y>=0,y+(a-b)*x^2+(2*a-3*b+c)*a*x+c-b<=0",17);pp
 *** using Lazard's method (BM20).
[y,2]
1
[x,1]
0
[c,2]
[b,1]
[a,2]
[[[a],[b],[],[],[]]]
time = 680 ms.
7 21(0,0) 87(20,18) 335(28,0) 434(0,274) 
 *** combined adjacent 214 cells.
time = 118 ms.
%2 = [[a,2*a^2-1],[a-b],[4*a^4+(-12*b+4*c)*a^3+(9*b^2-6*c*b+c^2)*a^2+(4*b-4*c)*a+(-4*b^2+4*c*b),b-c],[(a-b)*x^2+(2*a^2+(-3*b+c)*a)*x+(-b+c)],[(a-b)*x^2+(2*a^2+(-3*b+c)*a)*x+(y+(-b+c)),y]]

? tst12([],[a,b,c,x,y],(f1,f2)->f1*f2,"y>=0,y+(a-b)*x^2+(2*a-3*b+c)*a*x+c-b<=0",7);pp
 *** using the sum of squares projection.
[y,2]
[x,1]
[c,2]
[b,2]
[a,4]
time = 163 ms.
7 21(2,2) 63(34,14) 215(28,0) 278(0,178) 
 *** combined adjacent 142 cells.
time = 85 ms.
%3 = [[a,2*a^2-1,5*a^2-6*a+2,5*a^2+6*a+2],[a-b,16*a^8-96*b*a^7+(216*b^2+4)*a^6+(-216*b^3+32*b)*a^5+(81*b^4-128*b^2-8)*a^4+(168*b^3+8*b)*a^3+(-72*b^4+16*b^2+5)*a^2+(-32*b^3-8*b)*a+(16*b^4+4*b^2)],[10*a^4+(-38*b+8*c)*a^3+(53*b^2-20*c*b+2*c^2)*a^2+(-34*b^3+16*c*b^2-2*c^2*b)*a+(10*b^4-6*c*b^3+(c^2+1)*b^2-2*c*b+c^2),4*a^4+(-12*b+4*c)*a^3+(9*b^2-6*c*b+c^2)*a^2+(4*b-4*c)*a+(-4*b^2+4*c*b)],[(a-b)*x^2+(2*a^2+(-3*b+c)*a)*x+(-b+c)],[y,(a-b)*x^2+(2*a^2+(-3*b+c)*a)*x+(y+(-b+c))]]

C.W.Brown, S.McCallum "Enhancements to Lazard’s Method for Cylindrical Algebraic Decomposition"