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D.2.4.8 locus

Procedure from library grobcov.lib (see grobcov_lib).

Usage:
locus(G);
The input must be the grobcov of a parametrical ideal RETURN: The locus.
The output are the components of four constructible subsets of the locus of the parametrical system: Normal , Special, Accumulation and Degenerate. These components are
given as a list of (pi,(pi1,..pis_i),type_i,level_i) varying i, where the p's are prime ideals, the type can be: Normal, Special, Accumulation, Degenerate.
NOTE: It can only be called after computing the grobcov of the parametrical ideal in generic representation ('ext',0), which is the default.
The basering R, must be of the form Q[a_1,..,a_m][x_1,..,x_n]. KEYWORDS: geometrical locus, locus, loci.
EXAMPLE: locus; shows an example

Example:
 
LIB "grobcov.lib";
ring R=(0,a,b),(x4,x3,x2,x1),dp;
ideal S=(x1-3)^2+(x2-1)^2-9,
(4-x2)*(x3-3)+(x1-3)*(x4-1),
(3-x1)*(x3-x1)+(4-x2)*(x4-x2),
(4-x4)*a+(x3-3)*b+3*x4-4*x3,
(a-x1)^2+(b-x2)^2-(x1-x3)^2-(x2-x4)^2;
short=0;
locus(grobcov(S)); " ";
==> locus detected that the mover must avoid point (x1-3,x2-4) in order to ob\
   tain the correct locus
==> [1]:
==>    [1]:
==>       _[1]=(a^2-6*a+b^2+b+7)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Normal
==>    [4]:
==>       1
==> [2]:
==>    [1]:
==>       _[1]=(a^4-12*a^3+2*a^2*b^2-13*a^2*b+236*a^2-12*a*b^2+78*a*b-1200*a+\
   b^4-13*b^3+60*b^2-85*b+1495)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Normal
==>    [4]:
==>       1
==>  
kill R;
ring R=(0,a,b),(x,y),dp;
short=0;
"Concoid";
==> Concoid
ideal S96=x^2+y^2-4,(b-2)*x-a*y+2*a,(a-x)^2+(b-y)^2-1;
"System="; S96; " ";
==> System=
==> S96[1]=x^2+y^2-4
==> S96[2]=(b-2)*x+(-a)*y+(2*a)
==> S96[3]=x^2+y^2+(-2*a)*x+(-2*b)*y+(a^2+b^2-1)
==>  
locus(grobcov(S96));
==> [1]:
==>    [1]:
==>       _[1]=(a^4+2*a^2*b^2-9*a^2+b^4-9*b^2+4*b+12)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Normal
==>    [4]:
==>       1
==> [2]:
==>    [1]:
==>       _[1]=(a^2+b^2-4*b+3)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Special
==>    [4]:
==>       1
kill R; ring R=(0,x,y),(x1,x2),dp;
ideal S=-(x - 5)*(x1 - 1) - (x2 - 2)*(y - 2),
(x1 - 5)^2 + (x2 - 2)^2 - 4,
-2*(x - 5)*(x2 - 2) + 2*(x1 - 5)*(y - 2);
"System="; S;
==> System=
==> S[1]=(-x+5)*x1+(-y+2)*x2+(x+2*y-9)
==> S[2]=x1^2+x2^2-10*x1-4*x2+25
==> S[3]=(2*y-4)*x1+(-2*x+10)*x2+(4*x-10*y)
locus(grobcov(S)); " ";
==> [1]:
==>    [1]:
==>       _[1]=(3*x^2-30*x-y^2+4*y+71)
==>    [2]:
==>       [1]:
==>          _[1]=(y-2)
==>          _[2]=(x-5)
==>    [3]:
==>       Normal
==>    [4]:
==>       1
==> [2]:
==>    [1]:
==>       _[1]=(y-2)
==>       _[2]=(x-5)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Accumulation
==>    [4]:
==>       1
==>  
ideal S1=-(x - x1)*(x1 - 4) + (x2 - y)*(x2 - 4),
(x1 - 4)^2 + (x2 - 2)^2 - 4,
-2*(x1 - 4)*(2*x2 - y - 4) - 2*(x - 2*x1 + 4)*(x2 - 2);
"System="; S1;
==> System=
==> S1[1]=x1^2+x2^2+(-x-4)*x1+(-y-4)*x2+(4*x+4*y)
==> S1[2]=x1^2+x2^2-8*x1-4*x2+16
==> S1[3]=(2*y)*x1+(-2*x+8)*x2+(4*x-8*y-16)
locus(grobcov(S1));
==> [1]:
==>    [1]:
==>       _[1]=(x-4)
==>    [2]:
==>       [1]:
==>          _[1]=(y)
==>          _[2]=(x-4)
==>    [3]:
==>       Special
==>    [4]:
==>       1
==> [2]:
==>    [1]:
==>       _[1]=(y)
==>       _[2]=(x-4)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Accumulation
==>    [4]:
==>       1


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