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D.7.1.33 orbit_variety
Procedure from library finvar.lib (see finvar_lib).
- Usage:
- orbit_variety(F,s);
F: a 1xm <matrix> defing an invariant ring, s: a <string> giving the
name for a new ring
- Return:
- a Groebner basis (type <ideal>, named G) for the ideal defining the
orbit variety (i.e. the syzygy ideal) in the new ring (named newring)
- Theory:
- The ideal of algebraic relations of the invariant ring generators is
calculated, then the variables of the original ring are eliminated and
the polynomials that are left over define the orbit variety
Example:
| | LIB "finvar.lib";
ring R=0,(x,y,z),dp;
matrix F[1][7]=x2+y2,z2,x4+y4,1,x2z-1y2z,xyz,x3y-1xy3;
orbit_variety(F,"");
print(G);
==> y(4)-1,
==> y(5)*y(6)-y(2)*y(7),
==> y(2)*y(3)-y(5)^2-2*y(6)^2,
==> y(1)^2*y(6)-2*y(3)*y(6)+y(5)*y(7),
==> y(1)^2*y(5)-y(3)*y(5)-2*y(6)*y(7),
==> y(1)^2*y(2)-y(2)*y(3)-2*y(6)^2,
==> y(1)^4-3*y(1)^2*y(3)+2*y(3)^2+2*y(7)^2
basering;
==> // coefficients: QQ
==> // number of vars : 7
==> // block 1 : ordering dp
==> // : names y(1) y(2) y(3) y(4) y(5) y(6) y(7)
==> // block 2 : ordering C
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