|
D.2.4.4 multigrobcov
Procedure from library grobcov.lib (see grobcov_lib).
- Return:
- The list whose first element is the generic case, and the
remaining elements are the grobcov over the different irreducible
components in the complementary of the generic segment.
the empty list if the generic case does not have basis 1.
- Note:
- The basering R, must be of the form Q[a][x], a=parameters,
x=variables, and should be defined previously. The ideal must
be defined on R.
Example:
| LIB "grobcov.lib";
"Generalization of the Steiner-Lehmus theorem";
==> Generalization of the Steiner-Lehmus theorem
ring R=(0,x,y),(a,b,m,n,p,r),lp;
ideal S=p^2-(x^2+y^2),
-a*(y)+b*(x+p),
-a*y+b*(x-1)+y,
(r-1)^2-((x-1)^2+y^2),
-m*(y)+n*(x+r-2) +y,
-m*y+n*x,
(a^2+b^2)-((m-1)^2+n^2);
short=0;
multigrobcov(S,list("can",0,"cgs",0,"comment",1));
==> Generic case =
==> [1]:
==> _[1]=1
==> [2]:
==> _[1]=1
==> [3]:
==> [1]:
==> _[1]=0
==> [2]:
==> [1]:
==> _[1]=(y)
==> [2]:
==> _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+84*x^6*\
y^4+246*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278*x^4*y\
^4+31*x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+122*x^2\
*y^6+14*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8-y^6-6\
*y^4+y^2)
==> [3]:
==> _[1]=(2*x-1)
==> [4]:
==> _[1]=(16*x^11*y-88*x^10*y+82*x^9*y^3+192*x^9*y-369*x^8*y^3-204*x^8*y+1\
68*x^7*y^5+656*x^7*y^3+96*x^7*y-588*x^6*y^5-574*x^6*y^3+172*x^5*y^7+808*x\
^5*y^5+226*x^5*y^3-16*x^5*y-430*x^4*y^7-550*x^4*y^5+9*x^4*y^3+4*x^4*y+88*\
x^3*y^9+416*x^3*y^7+164*x^3*y^5-40*x^3*y^3-132*x^2*y^9-194*x^2*y^7+10*x^2\
*y^5+10*x^2*y^3+18*x*y^11+72*x*y^9+34*x*y^7-24*x*y^5+2*x*y^3-9*y^11-14*y^\
9+y^7+6*y^5-y^3)
==>
==> Components to study=
==> [1]:
==> _[1]=(y)
==> [2]:
==> _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+84*x^6*y^4+24\
6*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278*x^4*y^4+31*\
x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+122*x^2*y^6+1\
4*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8-y^6-6*y^4+y\
^2)
==> [3]:
==> _[1]=(2*x-1)
==>
==> Begin grobcov on the variety N =
==> N[1]=(y)
==> Options: can = 0, extend = 1, cgs = 0, rep = 0
==> Number of segments in buildtree (total) = 3
==> Number of lpp segments in groupsegments = 3
==> Time in buildtree = 1 sec
==> Time in groupRtoPrep = 0 sec
==> Time in LCUnion + combine = 0 sec
==> Time in extend = 0 sec
==> Time for grobcov = 1 sec
==> Number of segments of grobcov = 3
==>
==> Begin grobcov on the variety N =
==> N[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+84*x^6*y^4+246*x\
^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278*x^4*y^4+31*x^4\
*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+122*x^2*y^6+14*x\
^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8-y^6-6*y^4+y^2)
==> Options: can = 0, extend = 1, cgs = 0, rep = 0
==> Number of segments in buildtree (total) = 12
==> Number of lpp segments in groupsegments = 12
==> Time in buildtree = 11 sec
==> Time in groupRtoPrep = 186 sec
==> Time in LCUnion + combine = 0 sec
==> Time in extend = 64 sec
==> Time for grobcov = 262 sec
==> Number of segments of grobcov = 12
==>
==> Begin grobcov on the variety N =
==> N[1]=(2*x-1)
==> Options: can = 0, extend = 1, cgs = 0, rep = 0
==> Number of segments in buildtree (total) = 5
==> Number of lpp segments in groupsegments = 5
==> Time in buildtree = 2 sec
==> Time in groupRtoPrep = 0 sec
==> Time in LCUnion + combine = 0 sec
==> Time in extend = 0 sec
==> Time for grobcov = 3 sec
==> Number of segments of grobcov = 5
==> Time for multigrobcov = 267
==> [1]:
==> [1]:
==> [1]:
==> _[1]=1
==> [2]:
==> _[1]=1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=0
==> [2]:
==> [1]:
==> _[1]=(y)
==> [2]:
==> _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^\
7+84*x^6*y^4+246*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+\
278*x^4*y^4+31*x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^\
8+122*x^2*y^6+14*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*\
y^8-y^6-6*y^4+y^2)
==> [3]:
==> _[1]=(2*x-1)
==> [2]:
==> [1]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=b
==> _[5]=a^2
==> [2]:
==> _[1]=r^2-2*r+(-x^2+2*x)
==> _[2]=p^2+(-x^2)
==> _[3]=n
==> _[4]=b
==> _[5]=a^2-m^2+2*m-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> [2]:
==> [1]:
==> _[1]=(y)
==> _[2]=(x-1)
==> [2]:
==> _[1]=(y)
==> _[2]=(x)
==> [2]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n*r
==> _[4]=b
==> _[5]=a^2
==> [2]:
==> _[1]=r^2-2*r
==> _[2]=p^2
==> _[3]=n*r-2*n
==> _[4]=b
==> _[5]=a^2-m^2+2*m-n^2-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> _[2]=(x)
==> [2]:
==> [1]:
==> _[1]=1
==> [3]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=b*p
==> _[5]=a^2
==> [2]:
==> _[1]=r^2-2*r+1
==> _[2]=p^2-1
==> _[3]=n
==> _[4]=b*p+b
==> _[5]=a^2+b^2-m^2+2*m-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> _[2]=(x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=r
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=(3*x^4-6*x^3+6*x^2*y^2+5*x^2-6*x*y^2+3*y^4+5*y^2-1)*r+(x^5-\
10*x^4+2*x^3*y^2+17*x^3-18*x^2*y^2-10*x^2+x*y^4+17*x*y^2-x-8*y^4-10*y^2+2\
)
==> _[2]=(3*x^4-6*x^3+6*x^2*y^2+5*x^2-6*x*y^2-4*x+3*y^4+5*y^2+1)*p+(\
x^5+2*x^4+2*x^3*y^2-7*x^3+6*x^2*y^2+4*x^2+x*y^4-7*x*y^2-x+4*y^4+4*y^2)
==> _[3]=(x^5-4*x^4+2*x^3*y^2+5*x^3-6*x^2*y^2+x*y^4+5*x*y^2-x-2*y^4)\
*n+(-3*x^4*y+6*x^3*y-6*x^2*y^3-5*x^2*y+6*x*y^3-3*y^5-5*y^3+y)
==> _[4]=(x^5-4*x^4+2*x^3*y^2+5*x^3-6*x^2*y^2+x*y^4+5*x*y^2-x-2*y^4)\
*m+(-3*x^5+6*x^4-6*x^3*y^2-5*x^3+6*x^2*y^2-3*x*y^4-5*x*y^2+x)
==> _[5]=(x^5-x^4+2*x^3*y^2-x^3-x^2+x*y^4-x*y^2+3*x+y^4-y^2-1)*b+(3*\
x^4*y-6*x^3*y+6*x^2*y^3+5*x^2*y-6*x*y^3-4*x*y+3*y^5+5*y^3+y)
==> _[6]=(x^5-x^4+2*x^3*y^2-x^3-x^2+x*y^4-x*y^2+3*x+y^4-y^2-1)*a+(2*\
x^5-8*x^4+4*x^3*y^2+12*x^3-12*x^2*y^2-8*x^2+2*x*y^4+12*x*y^2+2*x-4*y^4-4*\
y^2)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+8\
4*x^6*y^4+246*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278\
*x^4*y^4+31*x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+1\
22*x^2*y^6+14*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8\
-y^6-6*y^4+y^2)
==> [2]:
==> [1]:
==> _[1]=(y)
==> _[2]=(x-1)
==> [2]:
==> _[1]=(y)
==> _[2]=(x)
==> [3]:
==> _[1]=(y)
==> _[2]=(2*x^2-2*x-1)
==> [4]:
==> _[1]=(4*y^2-3)
==> _[2]=(2*x-1)
==> [5]:
==> _[1]=(12*y^2-1)
==> _[2]=(2*x-1)
==> [6]:
==> _[1]=(y^4+11*y^2-1)
==> _[2]=(5*x+2*y^2+1)
==> [7]:
==> _[1]=(y^4+11*y^2-1)
==> _[2]=(5*x-2*y^2-6)
==> [8]:
==> _[1]=(4*y^4+5*y^2+2)
==> _[2]=(2*x-1)
==> [9]:
==> _[1]=(y^2+2*y-1)
==> _[2]=(x^2-x-2*y+1)
==> [10]:
==> _[1]=(y^2-2*y-1)
==> _[2]=(x^2-x+2*y+1)
==> [11]:
==> _[1]=(4*y^4+349*y^2-64)
==> _[2]=(17*x-2*y^2-15)
==> [12]:
==> _[1]=(4*y^4+349*y^2-64)
==> _[2]=(17*x+2*y^2-2)
==> [2]:
==> [1]:
==> _[1]=r
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=(2*x)*r+(-4*x-y^2+1)
==> _[2]=(2*x-2)*p+(2*x-y^2-1)
==> _[3]=(y^2-1)*n+(2*x*y)
==> _[4]=(2*x)*m+(-4*x-y^2+1)
==> _[5]=(y^2-1)*b+(-2*x*y+2*y)
==> _[6]=(2*x-2)*a+(2*x-y^2-1)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y^2-2*y-1)
==> _[2]=(x^2-x+2*y+1)
==> [2]:
==> [1]:
==> _[1]=1
==> [2]:
==> [1]:
==> _[1]=(y^2+2*y-1)
==> _[2]=(x^2-x-2*y+1)
==> [2]:
==> [1]:
==> _[1]=1
==> [3]:
==> [1]:
==> _[1]=r^2
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=5*r^2-10*r+(-y^2-3)
==> _[2]=5*p+(-y^2+2)
==> _[3]=(5*y)*n+(-3*y^2+1)*r
==> _[4]=5*m+(y^2-2)*r
==> _[5]=(5*y)*b+(3*y^2-1)
==> _[6]=a+1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y^4+11*y^2-1)
==> _[2]=(5*x+2*y^2+1)
==> [2]:
==> [1]:
==> _[1]=1
==> [4]:
==> [1]:
==> _[1]=r^2
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=4*r^2-8*r+(-4*y^2+3)
==> _[2]=p+r-1
==> _[3]=(8*y)*n+(-3*y^2+2)*r
==> _[4]=4*m+(-y^2-2)*r
==> _[5]=(8*y)*b+(-3*y^2+2)*r
==> _[6]=4*a+(y^2+2)*r-4
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(4*y^4+5*y^2+2)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [5]:
==> [1]:
==> _[1]=r
==> _[2]=p^2
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=5*r+(y^2-7)
==> _[2]=5*p^2+(-y^2-8)
==> _[3]=(5*y)*n+(3*y^2-1)
==> _[4]=m-2
==> _[5]=(5*y)*b+(3*y^2-1)*p+(-3*y^2+1)
==> _[6]=5*a+(y^2-2)*p+(-y^2-3)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y^4+11*y^2-1)
==> _[2]=(5*x-2*y^2-6)
==> [2]:
==> [1]:
==> _[1]=1
==> [6]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=3*r^2-6*r+2
==> _[2]=3*p^2-1
==> _[3]=2*n+(-3*y)*r
==> _[4]=4*m-3*r
==> _[5]=2*b+(3*y)*p+(-3*y)
==> _[6]=4*a-3*p-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(12*y^2-1)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [7]:
==> [1]:
==> _[1]=r
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=85*r+(4*y^2-123)
==> _[2]=p-1
==> _[3]=(170*y)*n+(47*y^2-64)
==> _[4]=85*m+(y^2-137)
==> _[5]=2*b+(-y)
==> _[6]=17*a+(-y^2-16)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(4*y^4+349*y^2-64)
==> _[2]=(17*x-2*y^2-15)
==> [2]:
==> [1]:
==> _[1]=1
==> [8]:
==> [1]:
==> _[1]=r
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=r
==> _[2]=85*p+(-4*y^2+38)
==> _[3]=2*n+(-y)
==> _[4]=17*m+(y^2-1)
==> _[5]=(170*y)*b+(47*y^2-64)
==> _[6]=85*a+(-y^2+52)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(4*y^4+349*y^2-64)
==> _[2]=(17*x+2*y^2-2)
==> [2]:
==> [1]:
==> _[1]=1
==> [9]:
==> [1]:
==> _[1]=r
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=r
==> _[2]=p-1
==> _[3]=2*n+(-y)
==> _[4]=4*m-1
==> _[5]=2*b+(-y)
==> _[6]=4*a-3
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(4*y^2-3)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [10]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=b
==> _[5]=a^2
==> [2]:
==> _[1]=2*r^2-4*r+(2*x-1)
==> _[2]=2*p^2+(-2*x-1)
==> _[3]=n
==> _[4]=b
==> _[5]=a^2-m^2+2*m-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> _[2]=(2*x^2-2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [11]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n*r
==> _[4]=b
==> _[5]=a^2
==> [2]:
==> _[1]=r^2-2*r
==> _[2]=p^2
==> _[3]=n*r-2*n
==> _[4]=b
==> _[5]=a^2-m^2+2*m-n^2-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> _[2]=(x)
==> [2]:
==> [1]:
==> _[1]=1
==> [12]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=b*p
==> _[5]=a^2
==> [2]:
==> _[1]=r^2-2*r+1
==> _[2]=p^2-1
==> _[3]=n
==> _[4]=b*p+b
==> _[5]=a^2+b^2-m^2+2*m-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> _[2]=(x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [4]:
==> [1]:
==> [1]:
==> _[1]=r^2
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=4*r^2-8*r+(-4*y^2+3)
==> _[2]=p+r-1
==> _[3]=(4*y^2-3)*n+(4*y)*r
==> _[4]=(4*y^2-3)*m+2*r
==> _[5]=(4*y^2-3)*b+(4*y)*r
==> _[6]=(4*y^2-3)*a-2*r+(-4*y^2+3)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=(y)
==> _[2]=(2*x-1)
==> [2]:
==> _[1]=(4*y^2-3)
==> _[2]=(2*x-1)
==> [3]:
==> _[1]=(12*y^2-1)
==> _[2]=(2*x-1)
==> [4]:
==> _[1]=(4*y^2+1)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=3*r^2-6*r+2
==> _[2]=3*p^2-1
==> _[3]=2*n+(-3*y)*r
==> _[4]=4*m-3*r
==> _[5]=2*b+(3*y)*p+(-3*y)
==> _[6]=4*a-3*p-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(12*y^2-1)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [3]:
==> [1]:
==> _[1]=r^2
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=r^2-2*r+1
==> _[2]=p+r-1
==> _[3]=n+(-y)*r
==> _[4]=2*m-r
==> _[5]=b+(-y)*r
==> _[6]=2*a+r-2
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(4*y^2+1)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [4]:
==> [1]:
==> _[1]=r
==> _[2]=p
==> _[3]=n
==> _[4]=m
==> _[5]=b
==> _[6]=a
==> [2]:
==> _[1]=r
==> _[2]=p-1
==> _[3]=2*n+(-y)
==> _[4]=4*m-1
==> _[5]=2*b+(-y)
==> _[6]=4*a-3
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(4*y^2-3)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
==> [5]:
==> [1]:
==> _[1]=r^2
==> _[2]=p^2
==> _[3]=n
==> _[4]=b
==> _[5]=a^2
==> [2]:
==> _[1]=4*r^2-8*r+3
==> _[2]=4*p^2-1
==> _[3]=n
==> _[4]=b
==> _[5]=a^2-m^2+2*m-1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(y)
==> _[2]=(2*x-1)
==> [2]:
==> [1]:
==> _[1]=1
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