Singular Manual: parametrizeOrbit

| #### D.5.10.4 parametrizeOrbit Procedure from library `orbitparam.lib` (see orbitparam_lib). Usage: parametrizeOrbit(L,v); L list, v matrix. Assume: L is a list of strictly upper triangular n x n matrices of same size. The vector space <L> genererated by the elements of L should be closed under the Lie bracket. v is matrix of constants of size n x 1. The basering has at least size(L) variables. However we will only use tangentGens(L,v)[1] many of them. Return: list, with four entries - int, dimension of the orbit - matrix A over the basering giving a parametrization of the orbit of v under the action of exp(<L>). - list of integers, with the (row)-indices of entries which can be deleted by the action - the variables of the parametrization to solve for Theory: We apply the theorem of Chevalley-Rosenlicht. First we determine tangent space generators, then apply `matrixExp` to the generators, and finally take the product to obtain the parametrization. Example:
    LIB "orbitparam.lib";
    ring R = 0,(t(1..3)),dp;
    matrix L1[3][3] = 0,1,0, 0,0,0, 0,0,0;
    matrix L2[3][3] = 0,0,1, 0,0,0, 0,0,0;
    matrix L3[3][3] = 0,1,1, 0,0,1, 0,0,0;
    list L = L1,L2,L3;
    matrix v[3][1] = 1,2,3;
    parametrizeOrbit(L,v);
    ==> [1]:
    ==>    2
    ==> [2]:
    ==>    _[1,1]=1/6*t(1)^2+5/3*t(1)+t(2)+1
    ==>    _[2,1]=t(1)+2
    ==>    _[3,1]=3
    ==> [3]:
    ==>    [1]:
    ==>       2
    ==>    [2]:
    ==>       1
    ==> [4]:
    ==>    [1]:
    ==>       t(1)
    ==>    [2]:
    ==>       t(2)
    ring R1 = 0,(t(1..2)),dp;
    matrix L1[4][4] = 0,1,0,0, 0,0,0,0, 0,0,0,1, 0,0,0,0;
    matrix L2[4][4] = 0,0,1,0, 0,0,0,1, 0,0,0,0, 0,0,0,0;
    list L = L1,L2;
    matrix v[4][1] = 1,2,3,4;
    parametrizeOrbit(L,v);
    ==> [1]:
    ==>    2
    ==> [2]:
    ==>    _[1,1]=1/4*t(1)*t(2)+1/2*t(1)+3/4*t(2)+1
    ==>    _[2,1]=t(2)+2
    ==>    _[3,1]=t(1)+3
    ==>    _[4,1]=4
    ==> [3]:
    ==>    [1]:
    ==>       3
    ==>    [2]:
    ==>       2
    ==> [4]:
    ==>    [1]:
    ==>       t(1)
    ==>    [2]:
    ==>       t(2)

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