? tst12([all,all,all],[a,b,c,d,e,f,x,y,z],id,"((a*x+b)*y+c)*z+((d*x+e)*y+f)<0");Ans()
*** using Lazard's method (MPP17).
[z,1]
[y,2]
[x,3]
[f,3]
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
time = 74 ms.
3 9(0,0) 27(0,0) 81(0,0) 291(0,60) 1257(0,396) 5800(0,1204) 23424(0,1720) 1(0,0)
*** combined adjacent 0 cells.
1[a = [a,1],b = [b,1],c = [c,1],d = [d,1],e = [e,1],f < [f,1],true,true,true]
time = 1,867 ms.
/* realrad version */
? tst12([all,all,all],[a,b,c,d,e,f,x,y,z],id,"((a*x+b)*y+c)*z+((d*x+e)*y+f)<0",17);Ans()
*** using Lazard's method (BM20).
6
[z,1]
2,2
[y,2]
0,5,0
[x,3]
1,1
[f,3]
1
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
time = 37,343 ms.
[[[],[],[],[d],[e],[],[],[],[]],[[a],[b],[],[],[],[],[],[],[]]]
3 9(0,0) 27(0,0) 81(0,0) 291(0,60) 1257(0,396) 5800(0,1204) 23424(0,1720) 1(0,0)
*** combined adjacent 0 cells.
1[a = [a,1],b = [b,1],c = [c,1],d = [d,1],e = [e,1],f < [f,1],true,true,true]
time = 1,882 ms.
/* recursive version */
? tst12([all,all,all],[a,b,c,d,e,f,x,y,z],id,"((a*x+b)*y+c)*z+((d*x+e)*y+f)<0",19);Ans()
1>> using Lazard's method (BM20).
[a, b, c, d, e, f, x, y],[d*y*x + (e*y + f), a*y*x + (b*y + c)]
2>> using Lazard's method (BM20).
[a, b, c, d, e, f, x],[f, d*x + e]
3>> using Lazard's method (BM20).
[a, b, c, d, e, f],[e, d]
4>> using Lazard's method (BM20).
[f]
[e,1]
[d,1]
[c]
[b]
[a]
<<4
time = 35 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
0
[x,1]
[f,1]
[e]
[d,1]
[c]
[b]
[a]
<<3
time = 111 ms.
1 1(0,0) 1(0,0) 3(0,0) 5(2,0) 5(0,0) 3(0,0)
*** combined adjacent 0 cells.
[a, b, c, d, e, f, x],[c, a*x + b]
3>> using Lazard's method (BM20).
[a, b, c, d, e, f],[b, a]
4>> using Lazard's method (BM20).
[f]
[e]
[d]
[c]
[b,1]
[a,1]
<<4
time = 34 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
0
[x,1]
[f]
[e]
[d]
[c,1]
[b]
[a,1]
<<3
time = 109 ms.
3 5(2,0) 5(0,0) 5(0,0) 5(0,0) 5(0,0) 3(0,0)
*** combined adjacent 0 cells.
1,1
[y,2]
[a, b, c, d, e, f],[e, d]
3>> using Lazard's method (BM20).
[f]
[e,1]
[d,1]
[c]
[b]
[a]
<<3
time = 37 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
[a, b, c, d, e, f],[-f*b + e*c, -f*a + d*c, -f*e*a + f*d*b]
3>> using Lazard's method (BM20).
[a, b, c, d, e],[b, e*c]
4>> using Lazard's method (BM20).
[e,1]
[d]
[c,1]
[b,1]
[a]
<<4
time = 46 ms.
1 1(0,0) 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
[a, b, c, d, e],[a, d*c]
4>> using Lazard's method (BM20).
[e]
[d,1]
[c,1]
[b]
[a,1]
<<4
time = 37 ms.
1 1(0,0) 3(0,0) 5(0,0) 5(0,0)
*** combined adjacent 2 cells.
1,1
[f,3]
[a, b, c, d],[a, d*b]
4>> using Lazard's method (BM20).
[d,1]
[c]
[b,1]
[a,1]
<<4
time = 42 ms.
1 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
1
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
<<3
time = 326 ms.
3 9(0,0) 27(0,0) 69(0,0) 227(0,60) 169(0,364)
*** combined adjacent 142 cells.
[a, b, c, d, e, f],[b, a]
3>> using Lazard's method (BM20).
[f]
[e]
[d]
[c]
[b,1]
[a,1]
<<3
time = 33 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
0,1,0
[x,3]
[a, b, c, d, e],[b, e*c]
3>> using Lazard's method (BM20).
[e,1]
[d]
[c,1]
[b,1]
[a]
<<3
time = 36 ms.
1 1(0,0) 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
[a, b, c, d, e],[a, d*c]
3>> using Lazard's method (BM20).
[e]
[d,1]
[c,1]
[b]
[a,1]
<<3
time = 33 ms.
1 1(0,0) 3(0,0) 5(0,0) 5(0,0)
*** combined adjacent 2 cells.
1,1
[f,3]
[a, b, c, d],[a, d*b]
3>> using Lazard's method (BM20).
[d,1]
[c]
[b,1]
[a,1]
<<3
time = 34 ms.
1 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
1
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
<<2
time = 1,238 ms.
3 9(0,0) 27(0,0) 81(0,0) 291(0,60) 1257(0,396) 5801(0,1204) 1169(0,1496)
*** combined adjacent 476 cells.
1
[z,1]
[a, b, c, d, e, f, x],[f, d*x + e]
2>> using Lazard's method (BM20).
[a, b, c, d, e, f],[e, d]
3>> using Lazard's method (BM20).
[f]
[e,1]
[d,1]
[c]
[b]
[a]
<<3
time = 37 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
0
[x,1]
[f,1]
[e]
[d,1]
[c]
[b]
[a]
<<2
time = 120 ms.
1 1(0,0) 1(0,0) 3(0,0) 5(2,0) 5(0,0) 3(0,0)
*** combined adjacent 0 cells.
[a, b, c, d, e, f, x],[c, a*x + b]
2>> using Lazard's method (BM20).
[a, b, c, d, e, f],[b, a]
3>> using Lazard's method (BM20).
[f]
[e]
[d]
[c]
[b,1]
[a,1]
<<3
time = 30 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
0
[x,1]
[f]
[e]
[d]
[c,1]
[b]
[a,1]
<<2
time = 113 ms.
3 5(2,0) 5(0,0) 5(0,0) 5(0,0) 5(0,0) 3(0,0)
*** combined adjacent 0 cells.
1,1
[y,2]
[a, b, c, d, e, f],[e, d]
2>> using Lazard's method (BM20).
[f]
[e,1]
[d,1]
[c]
[b]
[a]
<<2
time = 29 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
[a, b, c, d, e, f],[-f*b + e*c, -f*a + d*c, -f*e*a + f*d*b]
2>> using Lazard's method (BM20).
[a, b, c, d, e],[b, e*c]
3>> using Lazard's method (BM20).
[e,1]
[d]
[c,1]
[b,1]
[a]
<<3
time = 34 ms.
1 1(0,0) 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
[a, b, c, d, e],[a, d*c]
3>> using Lazard's method (BM20).
[e]
[d,1]
[c,1]
[b]
[a,1]
<<3
time = 40 ms.
1 1(0,0) 3(0,0) 5(0,0) 5(0,0)
*** combined adjacent 2 cells.
1,1
[f,3]
[a, b, c, d],[a, d*b]
3>> using Lazard's method (BM20).
[d,1]
[c]
[b,1]
[a,1]
<<3
time = 35 ms.
1 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
1
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
<<2
time = 312 ms.
3 9(0,0) 27(0,0) 69(0,0) 227(0,60) 169(0,364)
*** combined adjacent 142 cells.
[a, b, c, d, e, f],[b, a]
2>> using Lazard's method (BM20).
[f]
[e]
[d]
[c]
[b,1]
[a,1]
<<2
time = 35 ms.
1 1(0,0) 1(0,0) 1(0,0) 1(0,0) 1(0,0)
*** combined adjacent 0 cells.
0,1,0
[x,3]
[a, b, c, d, e],[b, e*c]
2>> using Lazard's method (BM20).
[e,1]
[d]
[c,1]
[b,1]
[a]
<<2
time = 39 ms.
1 1(0,0) 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
[a, b, c, d, e],[a, d*c]
2>> using Lazard's method (BM20).
[e]
[d,1]
[c,1]
[b]
[a,1]
<<2
time = 42 ms.
1 1(0,0) 3(0,0) 5(0,0) 5(0,0)
*** combined adjacent 2 cells.
1,1
[f,3]
[a, b, c, d],[a, d*b]
2>> using Lazard's method (BM20).
[d,1]
[c]
[b,1]
[a,1]
<<2
time = 40 ms.
1 3(0,0) 3(0,0) 5(0,0)
*** combined adjacent 2 cells.
1
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
<<1
time = 4,337 ms.
3 9(0,0) 27(0,0) 81(0,0) 291(0,60) 1257(0,396) 5800(0,1204) 23424(0,1720) 1(0,0)
*** combined adjacent 0 cells.
1[a = [a,1],b = [b,1],c = [c,1],d = [d,1],e = [e,1],f < [f,1],true,true,true]
time = 1,880 ms.
? tst12([all,all,all],[a,b,c,d,e,f,x,y,z],id,"((a*x+b)*y+c)*z+((d*x+e)*y+f)<0",7);Ans()
*** using the sum of squares projection.
[z,1]
[y,2]
[x,3]
[f,3]
[e,2]
[d,1]
[c,1]
[b,1]
[a,1]
time = 317 ms.
3 9(0,0) 27(0,0) 81(0,0) 291(0,60) 1097(272,386) 4808(0,1204) 13024(2000,1700) 1(0,0)
*** combined adjacent 0 cells.
1[a = [a,1],b = [b,1],c = [c,1],d = [d,1],e = [e,1],f < [f,1],true,true,true]
time = 1,724 ms.
Wolfram Language 12.2.0 Engine for Linux x86 (64-bit)
Copyright 1988-2021 Wolfram Research, Inc.
In[1]:= CylindricalDecomposition[((a*x+b)*y+c)*z+((d*x+e)*y+f)<0,{a,b,c,d,e,f,x,y,z},"Function"];//Timing
Out[1]= {9.409335, Null} (* The number of the top level cells is 5323. *)
In[2]:= SetSystemOptions["InequalitySolvingOptions"->{"BrownProjection" -> False, "LazardProjection" -> True}];
In[3]:= CylindricalDecomposition[((a*x+b)*y+c)*z+((d*x+e)*y+f)<0,{a,b,c,d,e,f,x,y,z},"Function"];//Timing
Out[3]= {14.773644, Null} (* The number of the top level cells is 4673. *)
