| #### 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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